Complete Coefficient Laws, Analytic Domains and Initial Boundary Terms
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batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations · 2026-10-15 · 100 questions
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1. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[1, 2, 3] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=0. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([1, 2, 3], a_n=0_after_the_list, 0, 1, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([1, 2, 3], a_n=0_after_the_list, 0, 1, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
2. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[2, 4, 6, 8] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=1. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([2, 4, 6, 8], a_n=0_after_the_list, 1, 4, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([2, 4, 6, 8], a_n=0_after_the_list, 1, 4, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
3. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[3, 6, 9, 12, 15] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=2. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([3, 6, 9, 12, 15], a_n=0_after_the_list, 2, 9, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15], a_n=0_after_the_list, 2, 9, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
4. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[4, 8, 12, 16, 20, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=3. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([4, 8, 12, 16, 20, 24], a_n=0_after_the_list, 3, 16, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24], a_n=0_after_the_list, 3, 16, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
5. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[5, 10, 15] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=4. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([5, 10, 15], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([5, 10, 15], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
6. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[6, 12, 18, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=5. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([6, 12, 18, 24], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
7. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[7, 14, 21, 28, 35] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=6. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([7, 14, 21, 28, 35], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
8. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[8, 16, 24, 32, 40, 48] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=7. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([8, 16, 24, 32, 40, 48], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
9. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[9, 18, 27] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=8. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([9, 18, 27], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([9, 18, 27], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
10. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[10, 20, 30, 40] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=0. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([10, 20, 30, 40], a_n=0_after_the_list, 0, 10, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([10, 20, 30, 40], a_n=0_after_the_list, 0, 10, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
11. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[11, 22, 33, 44, 55] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=1. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([11, 22, 33, 44, 55], a_n=0_after_the_list, 1, 22, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55], a_n=0_after_the_list, 1, 22, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
12. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[12, 24, 36, 48, 60, 72] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=2. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([12, 24, 36, 48, 60, 72], a_n=0_after_the_list, 2, 36, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72], a_n=0_after_the_list, 2, 36, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
13. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[13, 26, 39] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=3. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([13, 26, 39], a_n=0_after_the_list, 3, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([13, 26, 39], a_n=0_after_the_list, 3, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
14. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[14, 28, 42, 56] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=4. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([14, 28, 42, 56], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([14, 28, 42, 56], a_n=0_after_the_list, 4, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
15. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[15, 30, 45, 60, 75] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=5. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([15, 30, 45, 60, 75], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75], a_n=0_after_the_list, 5, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
16. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[16, 32, 48, 64, 80, 96] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=6. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([16, 32, 48, 64, 80, 96], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
17. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[17, 34, 51] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=7. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([17, 34, 51], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([17, 34, 51], a_n=0_after_the_list, 7, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
18. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[18, 36, 54, 72] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=8. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([18, 36, 54, 72], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([18, 36, 54, 72], a_n=0_after_the_list, 8, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
19. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[19, 38, 57, 76, 95] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=0. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([19, 38, 57, 76, 95], a_n=0_after_the_list, 0, 19, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95], a_n=0_after_the_list, 0, 19, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
20. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[20, 40, 60, 80, 100, 120] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=1. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([20, 40, 60, 80, 100, 120], a_n=0_after_the_list, 1, 40, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120], a_n=0_after_the_list, 1, 40, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
21. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[21, 42, 63] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=2. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([21, 42, 63], a_n=0_after_the_list, 2, 63, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([21, 42, 63], a_n=0_after_the_list, 2, 63, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
22. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[22, 44, 66, 88] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=3. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([22, 44, 66, 88], a_n=0_after_the_list, 3, 88, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([22, 44, 66, 88], a_n=0_after_the_list, 3, 88, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
23. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[23, 46, 69, 92, 115] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=4. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([23, 46, 69, 92, 115], a_n=0_after_the_list, 4, 115, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115], a_n=0_after_the_list, 4, 115, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
24. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[24, 48, 72, 96, 120, 144] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=5. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([24, 48, 72, 96, 120, 144], a_n=0_after_the_list, 5, 144, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144], a_n=0_after_the_list, 5, 144, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
25. Coefficient Extraction Uses the Declared Complete Zero Tail
Use finite_support_coefficients=[25, 50, 75] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Extract index=6. Give (finite_support_coefficients, zero_tail_law, index, coefficient, certificate).
View answer and explanation
([25, 50, 75], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law)
Return the indexed coefficient or0 beyond support. Ordinary generating coefficients are a_n, not a_n/n! of an exponential generating function. A finite list with an explicit zero tail differs from an unknown continuation. The exact result, retaining the requested order, is ([25, 50, 75], a_n=0_after_the_list, 6, 0, coefficient_extraction_uses_the_complete_finite_support_law).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-coefficient-extraction · candidate
26. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[1, 2, 3] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([1, 2, 3], 1/2, 2, [[0, 1, 1], [1, 2, 1], [2, 3, 3/4]], 11/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([1, 2, 3], 1/2, 2, [[0, 1, 1], [1, 2, 1], [2, 3, 3/4]], 11/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
27. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[2, 4, 6, 8] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([2, 4, 6, 8], 1/3, 2, [[0, 2, 2], [1, 4, 4/3], [2, 6, 2/3]], 4, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([2, 4, 6, 8], 1/3, 2, [[0, 2, 2], [1, 4, 4/3], [2, 6, 2/3]], 4, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
28. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[3, 6, 9, 12, 15] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([3, 6, 9, 12, 15], 1/4, 4, [[0, 3, 3], [1, 6, 3/2], [2, 9, 9/16], [3, 12, 3/16], [4, 15, 15/256]], 1359/256, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([3, 6, 9, 12, 15], 1/4, 4, [[0, 3, 3], [1, 6, 3/2], [2, 9, 9/16], [3, 12, 3/16], [4, 15, 15/256]], 1359/256, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
29. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[4, 8, 12, 16, 20, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([4, 8, 12, 16, 20, 24], 1/2, 4, [[0, 4, 4], [1, 8, 4], [2, 12, 3], [3, 16, 2], [4, 20, 5/4]], 57/4, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([4, 8, 12, 16, 20, 24], 1/2, 4, [[0, 4, 4], [1, 8, 4], [2, 12, 3], [3, 16, 2], [4, 20, 5/4]], 57/4, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
30. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[5, 10, 15] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([5, 10, 15], 1/3, 2, [[0, 5, 5], [1, 10, 10/3], [2, 15, 5/3]], 10, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([5, 10, 15], 1/3, 2, [[0, 5, 5], [1, 10, 10/3], [2, 15, 5/3]], 10, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
31. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[6, 12, 18, 24] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([6, 12, 18, 24], 1/4, 2, [[0, 6, 6], [1, 12, 3], [2, 18, 9/8]], 81/8, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([6, 12, 18, 24], 1/4, 2, [[0, 6, 6], [1, 12, 3], [2, 18, 9/8]], 81/8, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
32. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[7, 14, 21, 28, 35] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([7, 14, 21, 28, 35], 1/2, 4, [[0, 7, 7], [1, 14, 7], [2, 21, 21/4], [3, 28, 7/2], [4, 35, 35/16]], 399/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([7, 14, 21, 28, 35], 1/2, 4, [[0, 7, 7], [1, 14, 7], [2, 21, 21/4], [3, 28, 7/2], [4, 35, 35/16]], 399/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
33. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[8, 16, 24, 32, 40, 48] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([8, 16, 24, 32, 40, 48], 1/3, 4, [[0, 8, 8], [1, 16, 16/3], [2, 24, 8/3], [3, 32, 32/27], [4, 40, 40/81]], 1432/81, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
34. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[9, 18, 27] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([9, 18, 27], 1/4, 2, [[0, 9, 9], [1, 18, 9/2], [2, 27, 27/16]], 243/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([9, 18, 27], 1/4, 2, [[0, 9, 9], [1, 18, 9/2], [2, 27, 27/16]], 243/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
35. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[10, 20, 30, 40] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([10, 20, 30, 40], 1/2, 2, [[0, 10, 10], [1, 20, 10], [2, 30, 15/2]], 55/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([10, 20, 30, 40], 1/2, 2, [[0, 10, 10], [1, 20, 10], [2, 30, 15/2]], 55/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
36. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[11, 22, 33, 44, 55] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([11, 22, 33, 44, 55], 1/3, 4, [[0, 11, 11], [1, 22, 22/3], [2, 33, 11/3], [3, 44, 44/27], [4, 55, 55/81]], 1969/81, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([11, 22, 33, 44, 55], 1/3, 4, [[0, 11, 11], [1, 22, 22/3], [2, 33, 11/3], [3, 44, 44/27], [4, 55, 55/81]], 1969/81, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
37. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[12, 24, 36, 48, 60, 72] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([12, 24, 36, 48, 60, 72], 1/4, 4, [[0, 12, 12], [1, 24, 6], [2, 36, 9/4], [3, 48, 3/4], [4, 60, 15/64]], 1359/64, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([12, 24, 36, 48, 60, 72], 1/4, 4, [[0, 12, 12], [1, 24, 6], [2, 36, 9/4], [3, 48, 3/4], [4, 60, 15/64]], 1359/64, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
38. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[13, 26, 39] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([13, 26, 39], 1/2, 2, [[0, 13, 13], [1, 26, 13], [2, 39, 39/4]], 143/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([13, 26, 39], 1/2, 2, [[0, 13, 13], [1, 26, 13], [2, 39, 39/4]], 143/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
39. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[14, 28, 42, 56] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([14, 28, 42, 56], 1/3, 2, [[0, 14, 14], [1, 28, 28/3], [2, 42, 14/3]], 28, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([14, 28, 42, 56], 1/3, 2, [[0, 14, 14], [1, 28, 28/3], [2, 42, 14/3]], 28, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
40. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[15, 30, 45, 60, 75] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([15, 30, 45, 60, 75], 1/4, 4, [[0, 15, 15], [1, 30, 15/2], [2, 45, 45/16], [3, 60, 15/16], [4, 75, 75/256]], 6795/256, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([15, 30, 45, 60, 75], 1/4, 4, [[0, 15, 15], [1, 30, 15/2], [2, 45, 45/16], [3, 60, 15/16], [4, 75, 75/256]], 6795/256, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
41. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[16, 32, 48, 64, 80, 96] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([16, 32, 48, 64, 80, 96], 1/2, 4, [[0, 16, 16], [1, 32, 16], [2, 48, 12], [3, 64, 8], [4, 80, 5]], 57, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([16, 32, 48, 64, 80, 96], 1/2, 4, [[0, 16, 16], [1, 32, 16], [2, 48, 12], [3, 64, 8], [4, 80, 5]], 57, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
42. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[17, 34, 51] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([17, 34, 51], 1/3, 2, [[0, 17, 17], [1, 34, 34/3], [2, 51, 17/3]], 34, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([17, 34, 51], 1/3, 2, [[0, 17, 17], [1, 34, 34/3], [2, 51, 17/3]], 34, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
43. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[18, 36, 54, 72] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([18, 36, 54, 72], 1/4, 2, [[0, 18, 18], [1, 36, 9], [2, 54, 27/8]], 243/8, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([18, 36, 54, 72], 1/4, 2, [[0, 18, 18], [1, 36, 9], [2, 54, 27/8]], 243/8, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
44. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[19, 38, 57, 76, 95] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([19, 38, 57, 76, 95], 1/2, 4, [[0, 19, 19], [1, 38, 19], [2, 57, 57/4], [3, 76, 19/2], [4, 95, 95/16]], 1083/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([19, 38, 57, 76, 95], 1/2, 4, [[0, 19, 19], [1, 38, 19], [2, 57, 57/4], [3, 76, 19/2], [4, 95, 95/16]], 1083/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
45. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[20, 40, 60, 80, 100, 120] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([20, 40, 60, 80, 100, 120], 1/3, 4, [[0, 20, 20], [1, 40, 40/3], [2, 60, 20/3], [3, 80, 80/27], [4, 100, 100/81]], 3580/81, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([20, 40, 60, 80, 100, 120], 1/3, 4, [[0, 20, 20], [1, 40, 40/3], [2, 60, 20/3], [3, 80, 80/27], [4, 100, 100/81]], 3580/81, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
46. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[21, 42, 63] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([21, 42, 63], 1/4, 2, [[0, 21, 21], [1, 42, 21/2], [2, 63, 63/16]], 567/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([21, 42, 63], 1/4, 2, [[0, 21, 21], [1, 42, 21/2], [2, 63, 63/16]], 567/16, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
47. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[22, 44, 66, 88] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([22, 44, 66, 88], 1/2, 2, [[0, 22, 22], [1, 44, 22], [2, 66, 33/2]], 121/2, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
48. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[23, 46, 69, 92, 115] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/3, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([23, 46, 69, 92, 115], 1/3, 4, [[0, 23, 23], [1, 46, 46/3], [2, 69, 23/3], [3, 92, 92/27], [4, 115, 115/81]], 4117/81, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([23, 46, 69, 92, 115], 1/3, 4, [[0, 23, 23], [1, 46, 46/3], [2, 69, 23/3], [3, 92, 92/27], [4, 115, 115/81]], 4117/81, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
49. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[24, 48, 72, 96, 120, 144] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/4, cutoff=4. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([24, 48, 72, 96, 120, 144], 1/4, 4, [[0, 24, 24], [1, 48, 12], [2, 72, 9/2], [3, 96, 3/2], [4, 120, 15/32]], 1359/32, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([24, 48, 72, 96, 120, 144], 1/4, 4, [[0, 24, 24], [1, 48, 12], [2, 72, 9/2], [3, 96, 3/2], [4, 120, 15/32]], 1359/32, No, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
50. Support Coverage Is Separate from Accidental Equality at One Point
Use finite_support_coefficients=[25, 50, 75] in increasing index order and the complete law a_n=0 after the list. Canonical answers trim trailing zero coefficients, without changing the original shared parameters. This is a complete finite-support law, not an unknown sequence observed only through a prefix. Use x=1/2, cutoff=2. Give (finite_support_coefficients, x, cutoff, term_rows, finite_sum, complete_support_included, certificate).
complete_support_included certifies coverage of all nonzero support, not accidental equality at one x. At x=0 values may coincide before all support is included.
View answer and explanation
([25, 50, 75], 1/2, 2, [[0, 25, 25], [1, 50, 25], [2, 75, 75/4]], 275/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included)
List [n,a_n,a_n*x^n] through the cutoff and sum exactly. complete_support_included means every nonzero-support coefficient is included; it is not a pointwise-equality test. At x=0 or through cancellation a shorter sum may have the same value without being the whole polynomial identity. The exact result, retaining the requested order, is ([25, 50, 75], 1/2, 2, [[0, 25, 25], [1, 50, 25], [2, 75, 75/4]], 275/4, Yes, finite_partial_sum_equals_the_full_OGF_only_after_all_nonzero_finite_support_coefficients_are_included).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-finite-partial-sum · candidate
51. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(1/2)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(1/2)^n_for_every_n>=0, 1/2, 1/4, 4/3, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(1/2)^n_for_every_n>=0, 1/2, 1/4, 4/3, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
52. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(2/3)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(2/3)^n_for_every_n>=0, 1/2, 1/3, 3/2, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(2/3)^n_for_every_n>=0, 1/2, 1/3, 3/2, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
53. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(3/4)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(3/4)^n_for_every_n>=0, 1/2, 3/8, 8/5, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(3/4)^n_for_every_n>=0, 1/2, 3/8, 8/5, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
54. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(4/5)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(4/5)^n_for_every_n>=0, 1/2, 2/5, 5/3, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(4/5)^n_for_every_n>=0, 1/2, 2/5, 5/3, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
55. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(5/6)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(5/6)^n_for_every_n>=0, 1/2, 5/12, 12/7, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(5/6)^n_for_every_n>=0, 1/2, 5/12, 12/7, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
56. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(6/7)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(6/7)^n_for_every_n>=0, 1/2, 3/7, 7/4, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(6/7)^n_for_every_n>=0, 1/2, 3/7, 7/4, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
57. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(7/8)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(7/8)^n_for_every_n>=0, 1/2, 7/16, 16/9, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(7/8)^n_for_every_n>=0, 1/2, 7/16, 16/9, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
58. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(8/9)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(8/9)^n_for_every_n>=0, 1/2, 4/9, 9/5, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(8/9)^n_for_every_n>=0, 1/2, 4/9, 9/5, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
59. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(9/10)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(9/10)^n_for_every_n>=0, 1/2, 9/20, 20/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
60. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(10/11)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(10/11)^n_for_every_n>=0, 1/2, 5/11, 11/6, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(10/11)^n_for_every_n>=0, 1/2, 5/11, 11/6, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
61. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(11/12)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(11/12)^n_for_every_n>=0, 1/2, 11/24, 24/13, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(11/12)^n_for_every_n>=0, 1/2, 11/24, 24/13, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
62. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(12/13)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(12/13)^n_for_every_n>=0, 1/2, 6/13, 13/7, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(12/13)^n_for_every_n>=0, 1/2, 6/13, 13/7, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
63. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(13/14)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(13/14)^n_for_every_n>=0, 1/2, 13/28, 28/15, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(13/14)^n_for_every_n>=0, 1/2, 13/28, 28/15, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
64. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(14/15)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(14/15)^n_for_every_n>=0, 1/2, 7/15, 15/8, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(14/15)^n_for_every_n>=0, 1/2, 7/15, 15/8, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
65. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(15/16)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(15/16)^n_for_every_n>=0, 1/2, 15/32, 32/17, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(15/16)^n_for_every_n>=0, 1/2, 15/32, 32/17, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
66. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(16/17)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(16/17)^n_for_every_n>=0, 1/2, 8/17, 17/9, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(16/17)^n_for_every_n>=0, 1/2, 8/17, 17/9, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
67. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(17/18)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(17/18)^n_for_every_n>=0, 1/2, 17/36, 36/19, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(17/18)^n_for_every_n>=0, 1/2, 17/36, 36/19, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
68. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(18/19)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(18/19)^n_for_every_n>=0, 1/2, 9/19, 19/10, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(18/19)^n_for_every_n>=0, 1/2, 9/19, 19/10, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
69. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(19/20)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(19/20)^n_for_every_n>=0, 1/2, 19/40, 40/21, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(19/20)^n_for_every_n>=0, 1/2, 19/40, 40/21, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
70. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(20/21)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(20/21)^n_for_every_n>=0, 1/2, 10/21, 21/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(20/21)^n_for_every_n>=0, 1/2, 10/21, 21/11, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
71. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(21/22)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(21/22)^n_for_every_n>=0, 1/2, 21/44, 44/23, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(21/22)^n_for_every_n>=0, 1/2, 21/44, 44/23, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
72. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(22/23)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(22/23)^n_for_every_n>=0, 1/2, 11/23, 23/12, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(22/23)^n_for_every_n>=0, 1/2, 11/23, 23/12, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
73. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(23/24)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(23/24)^n_for_every_n>=0, 1/2, 23/48, 48/25, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(23/24)^n_for_every_n>=0, 1/2, 23/48, 48/25, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
74. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(24/25)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(24/25)^n_for_every_n>=0, 1/2, 12/25, 25/13, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(24/25)^n_for_every_n>=0, 1/2, 12/25, 25/13, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
75. The Geometric Analytic Closed Form Requires Absolute r Times x Less Than One
Use a_n=(25/26)^n for every n≥0 and evaluation x=1/2. Give (coefficient_law, x, ratio_times_x, analytic_value, domain_certificate, certificate).
View answer and explanation
(a_n=(25/26)^n_for_every_n>=0, 1/2, 25/52, 52/27, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one)
Require |r*x|<1 before asserting A(x)=1/(1−r*x). A finite denominator alone does not justify analytic equality outside convergence; formal power-series algebra has a separate scope. The exact result, retaining the requested order, is (a_n=(25/26)^n_for_every_n>=0, 1/2, 25/52, 52/27, Yes, A(x)=1/(1-rx)_only_on_absolute_rx_less_than_one).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-geometric-closed-form · candidate
76. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=1, a_1=2, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(1, 2, [1, 2, 3, 5, 8, 13, 21, 34, 55, 89], 1/4, 5/4, 11/16, 20/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (1, 2, [1, 2, 3, 5, 8, 13, 21, 34, 55, 89], 1/4, 5/4, 11/16, 20/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
77. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=2, a_1=3, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(2, 3, [2, 3, 5, 8, 13, 21, 34, 55, 89, 144], 1/5, 11/5, 19/25, 55/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (2, 3, [2, 3, 5, 8, 13, 21, 34, 55, 89, 144], 1/5, 11/5, 19/25, 55/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
78. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=3, a_1=4, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(3, 4, [3, 4, 7, 11, 18, 29, 47, 76, 123, 199], 1/6, 19/6, 29/36, 114/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (3, 4, [3, 4, 7, 11, 18, 29, 47, 76, 123, 199], 1/6, 19/6, 29/36, 114/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
79. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=4, a_1=5, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(4, 5, [4, 5, 9, 14, 23, 37, 60, 97, 157, 254], 1/4, 17/4, 11/16, 68/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (4, 5, [4, 5, 9, 14, 23, 37, 60, 97, 157, 254], 1/4, 17/4, 11/16, 68/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
80. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=5, a_1=6, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(5, 6, [5, 6, 11, 17, 28, 45, 73, 118, 191, 309], 1/5, 26/5, 19/25, 130/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (5, 6, [5, 6, 11, 17, 28, 45, 73, 118, 191, 309], 1/5, 26/5, 19/25, 130/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
81. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=6, a_1=7, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(6, 7, [6, 7, 13, 20, 33, 53, 86, 139, 225, 364], 1/6, 37/6, 29/36, 222/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (6, 7, [6, 7, 13, 20, 33, 53, 86, 139, 225, 364], 1/6, 37/6, 29/36, 222/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
82. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=7, a_1=8, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(7, 8, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419], 1/4, 29/4, 11/16, 116/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (7, 8, [7, 8, 15, 23, 38, 61, 99, 160, 259, 419], 1/4, 29/4, 11/16, 116/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
83. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=8, a_1=9, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(8, 9, [8, 9, 17, 26, 43, 69, 112, 181, 293, 474], 1/5, 41/5, 19/25, 205/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (8, 9, [8, 9, 17, 26, 43, 69, 112, 181, 293, 474], 1/5, 41/5, 19/25, 205/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
84. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=9, a_1=10, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(9, 10, [9, 10, 19, 29, 48, 77, 125, 202, 327, 529], 1/6, 55/6, 29/36, 330/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (9, 10, [9, 10, 19, 29, 48, 77, 125, 202, 327, 529], 1/6, 55/6, 29/36, 330/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
85. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=10, a_1=11, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(10, 11, [10, 11, 21, 32, 53, 85, 138, 223, 361, 584], 1/4, 41/4, 11/16, 164/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (10, 11, [10, 11, 21, 32, 53, 85, 138, 223, 361, 584], 1/4, 41/4, 11/16, 164/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
86. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=11, a_1=12, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(11, 12, [11, 12, 23, 35, 58, 93, 151, 244, 395, 639], 1/5, 56/5, 19/25, 280/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (11, 12, [11, 12, 23, 35, 58, 93, 151, 244, 395, 639], 1/5, 56/5, 19/25, 280/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
87. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=12, a_1=13, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(12, 13, [12, 13, 25, 38, 63, 101, 164, 265, 429, 694], 1/6, 73/6, 29/36, 438/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (12, 13, [12, 13, 25, 38, 63, 101, 164, 265, 429, 694], 1/6, 73/6, 29/36, 438/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
88. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=13, a_1=14, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(13, 14, [13, 14, 27, 41, 68, 109, 177, 286, 463, 749], 1/4, 53/4, 11/16, 212/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (13, 14, [13, 14, 27, 41, 68, 109, 177, 286, 463, 749], 1/4, 53/4, 11/16, 212/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
89. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=14, a_1=15, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(14, 15, [14, 15, 29, 44, 73, 117, 190, 307, 497, 804], 1/5, 71/5, 19/25, 355/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (14, 15, [14, 15, 29, 44, 73, 117, 190, 307, 497, 804], 1/5, 71/5, 19/25, 355/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
90. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=15, a_1=16, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(15, 16, [15, 16, 31, 47, 78, 125, 203, 328, 531, 859], 1/6, 91/6, 29/36, 546/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (15, 16, [15, 16, 31, 47, 78, 125, 203, 328, 531, 859], 1/6, 91/6, 29/36, 546/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
91. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=16, a_1=17, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(16, 17, [16, 17, 33, 50, 83, 133, 216, 349, 565, 914], 1/4, 65/4, 11/16, 260/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (16, 17, [16, 17, 33, 50, 83, 133, 216, 349, 565, 914], 1/4, 65/4, 11/16, 260/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
92. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=17, a_1=18, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(17, 18, [17, 18, 35, 53, 88, 141, 229, 370, 599, 969], 1/5, 86/5, 19/25, 430/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (17, 18, [17, 18, 35, 53, 88, 141, 229, 370, 599, 969], 1/5, 86/5, 19/25, 430/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
93. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=18, a_1=19, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(18, 19, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024], 1/6, 109/6, 29/36, 654/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (18, 19, [18, 19, 37, 56, 93, 149, 242, 391, 633, 1024], 1/6, 109/6, 29/36, 654/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
94. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=19, a_1=20, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(19, 20, [19, 20, 39, 59, 98, 157, 255, 412, 667, 1079], 1/4, 77/4, 11/16, 28, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (19, 20, [19, 20, 39, 59, 98, 157, 255, 412, 667, 1079], 1/4, 77/4, 11/16, 28, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
95. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=20, a_1=21, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(20, 21, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134], 1/5, 101/5, 19/25, 505/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (20, 21, [20, 21, 41, 62, 103, 165, 268, 433, 701, 1134], 1/5, 101/5, 19/25, 505/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
96. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=21, a_1=22, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(21, 22, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189], 1/6, 127/6, 29/36, 762/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (21, 22, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189], 1/6, 127/6, 29/36, 762/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
97. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=22, a_1=23, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(22, 23, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244], 1/4, 89/4, 11/16, 356/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (22, 23, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244], 1/4, 89/4, 11/16, 356/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
98. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=23, a_1=24, and x=1/5 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (23, 24, [23, 24, 47, 71, 118, 189, 307, 496, 803, 1299], 1/5, 116/5, 19/25, 580/19, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
99. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=24, a_1=25, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(24, 25, [24, 25, 49, 74, 123, 197, 320, 517, 837, 1354], 1/6, 145/6, 29/36, 30, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (24, 25, [24, 25, 49, 74, 123, 197, 320, 517, 837, 1354], 1/6, 145/6, 29/36, 30, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
100. The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF
Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=25, a_1=26, and x=1/4 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).
View answer and explanation
(25, 26, [25, 26, 51, 77, 128, 205, 333, 538, 871, 1409], 1/4, 101/4, 11/16, 404/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)
The ten displayed coefficients are a preview of this complete law, not its definition. Retain numerator a0+(a1−a0)*x and denominator1−x−x^2. The declared domain is sufficient for analytic convergence, not an asserted exact radius for every initial pair. No unproved termwise differentiation or radius-outside substitution is inferred. The exact result, retaining the requested order, is (25, 26, [25, 26, 51, 77, 128, 205, 333, 538, 871, 1409], 1/4, 101/4, 11/16, 404/11, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved).
Reviewed step-by-step derivation
Analytic ordinary generating function · qf-mod44-ogf-recurrence-boundary-terms · candidate
Canonical question JSONL · English JSONL · Validation report