# Question 22562: Different Missing-Initial Completions Give Different Whole Prefixes [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-0087) 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. R0044-22562 · q-20261014-advance-20261110-r05-recurrence-relation-0087 ## Original question and givens Use order2 coefficients=[3, 3] oldest-to-newest, constant=3, known a_0=11, alternative a_1=12 or 13, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary). - G001 — Use order2 coefficients=[3, 3] oldest-to-newest, constant=3, known a_0=11, alternative a_1=12 or 13, target_n=8. Give (first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary). ## 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-missing-initial-boundary;original snapshot and mathematics retained After explicit ordered source output fields=[first_initial, second_left, second_right, left_prefix, right_prefix, prefixes_differ, one_initial_uniqueness_claim, boundary] 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 LEFT completion:ORIGINAL order=2;oldest-to-newest coefficients=[3, 3];constant=3;ALL initial a_0..a_(r-1)=[11, 12] After for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;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 LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 11) After recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 33 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 LEFT completion n=2 partial sum after coefficient0 := sum(3, 33) After recurrence LEFT completion n=2 partial sum after coefficient0 = 36 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 LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 12) After recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 36 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 LEFT completion n=2 partial sum after coefficient1 := sum(36, 36) After recurrence LEFT completion n=2 partial sum after coefficient1 = 72 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[11, 12];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 11, 33], [1, 1, 3, 12, 36]] After new a_2=72;next state=[12, 72];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72] 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, S006 S008 · calculationBefore recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 12) After recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 36 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 LEFT completion n=3 partial sum after coefficient0 := sum(3, 36) After recurrence LEFT completion n=3 partial sum after coefficient0 = 39 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 LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 72) After recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 216 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 · calculationBefore recurrence LEFT completion n=3 partial sum after coefficient1 := sum(39, 216) After recurrence LEFT completion n=3 partial sum after coefficient1 = 255 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, S010 S012 · impliesBefore recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[12, 72];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 12, 36], [1, 2, 3, 72, 216]] After new a_3=255;next state=[72, 255];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255] 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, S011 S013 · calculationBefore recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 72) After recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 216 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 LEFT completion n=4 partial sum after coefficient0 := sum(3, 216) After recurrence LEFT completion n=4 partial sum after coefficient0 = 219 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 LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 255) After recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 765 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 LEFT completion n=4 partial sum after coefficient1 := sum(219, 765) After recurrence LEFT completion n=4 partial sum after coefficient1 = 984 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[72, 255];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 72, 216], [1, 3, 3, 255, 765]] After new a_4=984;next state=[255, 984];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984] 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, S016 S018 · calculationBefore recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 255) After recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 765 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 LEFT completion n=5 partial sum after coefficient0 := sum(3, 765) After recurrence LEFT completion n=5 partial sum after coefficient0 = 768 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 · calculationBefore recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 984) After recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 2952 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, S019 S021 · calculationBefore recurrence LEFT completion n=5 partial sum after coefficient1 := sum(768, 2952) After recurrence LEFT completion n=5 partial sum after coefficient1 = 3720 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[255, 984];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 255, 765], [1, 4, 3, 984, 2952]] After new a_5=3720;next state=[984, 3720];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720] 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, S021 S023 · calculationBefore recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 984) After recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 2952 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 LEFT completion n=6 partial sum after coefficient0 := sum(3, 2952) After recurrence LEFT completion n=6 partial sum after coefficient0 = 2955 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 LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3720) After recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 11160 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 LEFT completion n=6 partial sum after coefficient1 := sum(2955, 11160) After recurrence LEFT completion n=6 partial sum after coefficient1 = 14115 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[984, 3720];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 984, 2952], [1, 5, 3, 3720, 11160]] After new a_6=14115;next state=[3720, 14115];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115] 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, S026 S028 · calculationBefore recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 3720) After recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 11160 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 · calculationBefore recurrence LEFT completion n=7 partial sum after coefficient0 := sum(3, 11160) After recurrence LEFT completion n=7 partial sum after coefficient0 = 11163 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, S028 S030 · calculationBefore recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 14115) After recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 42345 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 LEFT completion n=7 partial sum after coefficient1 := sum(11163, 42345) After recurrence LEFT completion n=7 partial sum after coefficient1 = 53508 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3720, 14115];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 3720, 11160], [1, 6, 3, 14115, 42345]] After new a_7=53508;next state=[14115, 53508];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115, 53508] 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, S031 S033 · calculationBefore recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 14115) After recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 42345 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 LEFT completion n=8 partial sum after coefficient0 := sum(3, 42345) After recurrence LEFT completion n=8 partial sum after coefficient0 = 42348 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 LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(3, 53508) After recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 160524 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 LEFT completion n=8 partial sum after coefficient1 := sum(42348, 160524) After recurrence LEFT completion n=8 partial sum after coefficient1 = 202872 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 · impliesBefore recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[14115, 53508];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 14115, 42345], [1, 7, 3, 53508, 160524]] After new a_8=202872;next state=[53508, 202872];drop ONLY oldest state entry then append new scalar;prefix now=[11, 12, 72, 255, 984, 3720, 14115, 53508, 202872] 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, S036 S038 · impliesBefore recurrence LEFT completion:complete ORIGINAL initial state=[11, 12];actual generated prefix=[11, 12, 72, 255, 984, 3720, 14115, 53508, 202872] After ANY two sequences with SAME r initials and SAME total rule agree at n=2 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, S037 S039 · impliesBefore recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[3, 3];constant=3;ALL initial a_0..a_(r-1)=[11, 13] After for EVERY n>=2,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=2;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, S038 S040 · calculationBefore recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 11) After recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 33 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 RIGHT completion n=2 partial sum after coefficient0 := sum(3, 33) After recurrence RIGHT completion n=2 partial sum after coefficient0 = 36 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 RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(3, 13) After recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 39 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 RIGHT completion n=2 partial sum after coefficient1 := sum(36, 39) After recurrence RIGHT completion n=2 partial sum after coefficient1 = 75 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 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[11, 13];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 11, 33], [1, 1, 3, 13, 39]] After new a_2=75;next state=[13, 75];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75] 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, S043 S045 · calculationBefore recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 13) After recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 39 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 RIGHT completion n=3 partial sum after coefficient0 := sum(3, 39) After recurrence RIGHT completion n=3 partial sum after coefficient0 = 42 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 · calculationBefore recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(3, 75) After recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 225 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, S046 S048 · calculationBefore recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(42, 225) After recurrence RIGHT completion n=3 partial sum after coefficient1 = 267 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 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[13, 75];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 13, 39], [1, 2, 3, 75, 225]] After new a_3=267;next state=[75, 267];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267] 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, S048 S050 · calculationBefore recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 75) After recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 225 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 RIGHT completion n=4 partial sum after coefficient0 := sum(3, 225) After recurrence RIGHT completion n=4 partial sum after coefficient0 = 228 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 RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(3, 267) After recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 801 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 RIGHT completion n=4 partial sum after coefficient1 := sum(228, 801) After recurrence RIGHT completion n=4 partial sum after coefficient1 = 1029 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 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[75, 267];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 75, 225], [1, 3, 3, 267, 801]] After new a_4=1029;next state=[267, 1029];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029] 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, S053 S055 · calculationBefore recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 267) After recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 801 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 · calculationBefore recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(3, 801) After recurrence RIGHT completion n=5 partial sum after coefficient0 = 804 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, S055 S057 · calculationBefore recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(3, 1029) After recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 3087 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, S056 S058 · calculationBefore recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(804, 3087) After recurrence RIGHT completion n=5 partial sum after coefficient1 = 3891 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, S057 S059 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[267, 1029];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 267, 801], [1, 4, 3, 1029, 3087]] After new a_5=3891;next state=[1029, 3891];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891] 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, S058 S060 · calculationBefore recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 1029) After recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 3087 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, S059 S061 · calculationBefore recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(3, 3087) After recurrence RIGHT completion n=6 partial sum after coefficient0 = 3090 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, S060 S062 · calculationBefore recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(3, 3891) After recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 11673 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, S061 S063 · calculationBefore recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(3090, 11673) After recurrence RIGHT completion n=6 partial sum after coefficient1 = 14763 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, S062 S064 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[1029, 3891];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 1029, 3087], [1, 5, 3, 3891, 11673]] After new a_6=14763;next state=[3891, 14763];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763] 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, S063 S065 · calculationBefore recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 3891) After recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 11673 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, S064 S066 · calculationBefore recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(3, 11673) After recurrence RIGHT completion n=7 partial sum after coefficient0 = 11676 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, S065 S067 · calculationBefore recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(3, 14763) After recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 44289 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, S066 S068 · calculationBefore recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(11676, 44289) After recurrence RIGHT completion n=7 partial sum after coefficient1 = 55965 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, S067 S069 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[3891, 14763];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 3891, 11673], [1, 6, 3, 14763, 44289]] After new a_7=55965;next state=[14763, 55965];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763, 55965] 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, S068 S070 · calculationBefore recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 14763) After recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 44289 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, S069 S071 · calculationBefore recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(3, 44289) After recurrence RIGHT completion n=8 partial sum after coefficient0 = 44292 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, S070 S072 · calculationBefore recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(3, 55965) After recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 167895 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, S071 S073 · calculationBefore recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(44292, 167895) After recurrence RIGHT completion n=8 partial sum after coefficient1 = 212187 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, S072 S074 · impliesBefore recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[14763, 55965];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 14763, 44289], [1, 7, 3, 55965, 167895]] After new a_8=212187;next state=[55965, 212187];drop ONLY oldest state entry then append new scalar;prefix now=[11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187] 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, S073 S075 · impliesBefore recurrence RIGHT completion:complete ORIGINAL initial state=[11, 13];actual generated prefix=[11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187] After ANY two sequences with SAME r initials and SAME total rule agree at n=2 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, S074 S076 · impliesBefore ORIGINAL only known a_0=11;different completions a_1=12 and13;target=8 After WHOLE prefixes including initials differ=Yes;actual targets=[202872, 212187] MAY coincide;one_initial_uniqueness_claim fixed No Conditions: none The prefixes already differ at a1,so the complete prefix is not unique;the target values need not remain different.The final No withholds order2 uniqueness from one initial,without negating prefixes_differ. Step identifiers and dependenciesRule: recurrence_missing_initial_WHOLE_prefix_not_target_difference · Uses: G001, S075 ## Reverse checks - S076 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S075 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S074 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S073 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 with exact rational arithmetic, not rounded decimals. - S071 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S070 · premise_and_result_audit — Recompute recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S068 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S067 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S063 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S058 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S057 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S053 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S048 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S043 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S036 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S031 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S026 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute recurrence LEFT completion n=2 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 (11, 12, 13, [11, 12, 72, 255, 984, 3720, 14115, 53508, 202872], [11, 13, 75, 267, 1029, 3891, 14763, 55965, 212187], Yes, No, order_two_recurrence_with_only_one_initial_value_is_not_unique) 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: 039d193b5ba093a65875133ff28299dad2c3fe41519d6ce1b4c8aaa2fafbbbc1 - candidate record SHA-256: 6f1ff696037a1694b02578f7b0631cc9cd7b1d7eaa10374e62f5dc3e1953c26a - 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-0087.json)