# Question 22540: A Complete Initial State Gives a Unique Forward Prefix [Return to catalogue page 113](https://ocme.evemisslab.com/en/questions/page/113/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261110-r05-recurrence-relation-foundations/#q-20261014-advance-20261110-r05-recurrence-relation-0065) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0044-22540 · q-20261014-advance-20261110-r05-recurrence-relation-0065 ## Original question and givens Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate). - G001 — Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-recurrence-relation](https://ocme.evemisslab.com/en/mko/mko-recurrence-relation/) ## Step-by-step derivation S001 · impliesBefore source operation=recurrence-initial-value-uniqueness;original snapshot and mathematics retained After explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=2;ALL initial a_0..a_(r-1)=[14, 15, 16, 17] After for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued Conditions: none Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim. Step identifiers and dependenciesRule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001 S003 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 14) After recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 14 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 14) After recurrence ORIGINAL n=4 partial sum after coefficient0 = 16 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 15) After recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 30 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(16, 30) After recurrence ORIGINAL n=4 partial sum after coefficient1 = 46 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 16) After recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 48 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(46, 48) After recurrence ORIGINAL n=4 partial sum after coefficient2 = 94 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 17) After recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 68 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(94, 68) After recurrence ORIGINAL n=4 partial sum after coefficient3 = 162 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[14, 15, 16, 17];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 14, 14], [1, 1, 2, 15, 30], [2, 2, 3, 16, 48], [3, 3, 4, 17, 68]] After new a_4=162;next state=[15, 16, 17, 162];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S010 S012 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 15) After recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 15 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 15) After recurrence ORIGINAL n=5 partial sum after coefficient0 = 17 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 16) After recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 32 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(17, 32) After recurrence ORIGINAL n=5 partial sum after coefficient1 = 49 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 17) After recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 51 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(49, 51) After recurrence ORIGINAL n=5 partial sum after coefficient2 = 100 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 162) After recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 648 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(100, 648) After recurrence ORIGINAL n=5 partial sum after coefficient3 = 748 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[15, 16, 17, 162];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 15, 15], [1, 2, 2, 16, 32], [2, 3, 3, 17, 51], [3, 4, 4, 162, 648]] After new a_5=748;next state=[16, 17, 162, 748];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S019 S021 · calculationBefore recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 16) After recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 16 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 16) After recurrence ORIGINAL n=6 partial sum after coefficient0 = 18 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 17) After recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 34 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(18, 34) After recurrence ORIGINAL n=6 partial sum after coefficient1 = 52 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 162) After recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 486 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · calculationBefore recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(52, 486) After recurrence ORIGINAL n=6 partial sum after coefficient2 = 538 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S025 S027 · calculationBefore recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 748) After recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 2992 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S026 S028 · calculationBefore recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(538, 2992) After recurrence ORIGINAL n=6 partial sum after coefficient3 = 3530 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S027 S029 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[16, 17, 162, 748];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 16, 16], [1, 3, 2, 17, 34], [2, 4, 3, 162, 486], [3, 5, 4, 748, 2992]] After new a_6=3530;next state=[17, 162, 748, 3530];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S028 S030 · calculationBefore recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 17) After recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 17 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S029 S031 · calculationBefore recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(2, 17) After recurrence ORIGINAL n=7 partial sum after coefficient0 = 19 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S030 S032 · calculationBefore recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 162) After recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 324 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S031 S033 · calculationBefore recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(19, 324) After recurrence ORIGINAL n=7 partial sum after coefficient1 = 343 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S032 S034 · calculationBefore recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 748) After recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 2244 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S033 S035 · calculationBefore recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(343, 2244) After recurrence ORIGINAL n=7 partial sum after coefficient2 = 2587 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S034 S036 · calculationBefore recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 3530) After recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 14120 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S035 S037 · calculationBefore recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(2587, 14120) After recurrence ORIGINAL n=7 partial sum after coefficient3 = 16707 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S036 S038 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[17, 162, 748, 3530];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 17, 17], [1, 4, 2, 162, 324], [2, 5, 3, 748, 2244], [3, 6, 4, 3530, 14120]] After new a_7=16707;next state=[162, 748, 3530, 16707];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S037 S039 · calculationBefore recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(1, 162) After recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 162 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S038 S040 · calculationBefore recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(2, 162) After recurrence ORIGINAL n=8 partial sum after coefficient0 = 164 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S039 S041 · calculationBefore recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(2, 748) After recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 1496 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S040 S042 · calculationBefore recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(164, 1496) After recurrence ORIGINAL n=8 partial sum after coefficient1 = 1660 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S041 S043 · calculationBefore recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(3, 3530) After recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 10590 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S042 S044 · calculationBefore recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(1660, 10590) After recurrence ORIGINAL n=8 partial sum after coefficient2 = 12250 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S043 S045 · calculationBefore recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(4, 16707) After recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 66828 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S044 S046 · calculationBefore recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(12250, 66828) After recurrence ORIGINAL n=8 partial sum after coefficient3 = 79078 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S045 S047 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[162, 748, 3530, 16707];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 162, 162], [1, 5, 2, 748, 1496], [2, 6, 3, 3530, 10590], [3, 7, 4, 16707, 66828]] After new a_8=79078;next state=[748, 3530, 16707, 79078];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S046 S048 · calculationBefore recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(1, 748) After recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 748 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S047 S049 · calculationBefore recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(2, 748) After recurrence ORIGINAL n=9 partial sum after coefficient0 = 750 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S048 S050 · calculationBefore recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(2, 3530) After recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 7060 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S049 S051 · calculationBefore recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(750, 7060) After recurrence ORIGINAL n=9 partial sum after coefficient1 = 7810 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S050 S052 · calculationBefore recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(3, 16707) After recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 50121 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S051 S053 · calculationBefore recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(7810, 50121) After recurrence ORIGINAL n=9 partial sum after coefficient2 = 57931 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S052 S054 · calculationBefore recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(4, 79078) After recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 316312 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S053 S055 · calculationBefore recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(57931, 316312) After recurrence ORIGINAL n=9 partial sum after coefficient3 = 374243 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S054 S056 · impliesBefore recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[748, 3530, 16707, 79078];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 748, 748], [1, 6, 2, 3530, 7060], [2, 7, 3, 16707, 50121], [3, 8, 4, 79078, 316312]] After new a_9=374243;next state=[3530, 16707, 79078, 374243];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243] Conditions: none Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update. Step identifiers and dependenciesRule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S055 S057 · impliesBefore recurrence ORIGINAL:complete ORIGINAL initial state=[14, 15, 16, 17];actual generated prefix=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243] After ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only Conditions: none Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity. Step identifiers and dependenciesRule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S056 ## Reverse checks - S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S055 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S046 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S037 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: dbc4b131fb556e5fb0f90934f539554b8ca1fa45bf989dfb79956217e82404f9 - candidate record SHA-256: 77fe653e125038a84c9c29ef4c786ca331576aea1eb2526f1bca2f50b27f7e61 - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261110-r05-recurrence-relation-0065.json)