Question 22553: Different Missing-Initial Completions Give Different Whole Prefixes

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0044-22553 · q-20261014-advance-20261110-r05-recurrence-relation-0078

Original question and givens

Use order2 coefficients=[3, 4] oldest-to-newest, constant=2, known a_0=2, alternative a_1=3 or 4, 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, 4] oldest-to-newest, constant=2, known a_0=2, alternative a_1=3 or 4, 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

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

recurrence LEFT completion:ORIGINAL order=2;oldest-to-newest coefficients=[3, 4];constant=2;ALL initial a_0..a_(r-1)=[2, 3]

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 dependencies

Rule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001

S003 · calculation

Before

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 2)

After

recurrence LEFT completion n=2 ORIGINAL coefficient0 times a_0 = 6

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

recurrence LEFT completion n=2 partial sum after coefficient0 := sum(2, 6)

After

recurrence LEFT completion n=2 partial sum after coefficient0 = 8

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 3)

After

recurrence LEFT completion n=2 ORIGINAL coefficient1 times a_1 = 12

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

recurrence LEFT completion n=2 partial sum after coefficient1 := sum(8, 12)

After

recurrence LEFT completion n=2 partial sum after coefficient1 = 20

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

recurrence LEFT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[2, 3];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 2, 6], [1, 1, 4, 3, 12]]

After

new a_2=20;next state=[3, 20];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20]

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 dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S006

S008 · calculation

Before

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 3)

After

recurrence LEFT completion n=3 ORIGINAL coefficient0 times a_1 = 9

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

recurrence LEFT completion n=3 partial sum after coefficient0 := sum(2, 9)

After

recurrence LEFT completion n=3 partial sum after coefficient0 = 11

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 20)

After

recurrence LEFT completion n=3 ORIGINAL coefficient1 times a_2 = 80

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

recurrence LEFT completion n=3 partial sum after coefficient1 := sum(11, 80)

After

recurrence LEFT completion n=3 partial sum after coefficient1 = 91

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

recurrence LEFT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[3, 20];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 3, 9], [1, 2, 4, 20, 80]]

After

new a_3=91;next state=[20, 91];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S011

S013 · calculation

Before

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 20)

After

recurrence LEFT completion n=4 ORIGINAL coefficient0 times a_2 = 60

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

recurrence LEFT completion n=4 partial sum after coefficient0 := sum(2, 60)

After

recurrence LEFT completion n=4 partial sum after coefficient0 = 62

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 91)

After

recurrence LEFT completion n=4 ORIGINAL coefficient1 times a_3 = 364

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

recurrence LEFT completion n=4 partial sum after coefficient1 := sum(62, 364)

After

recurrence LEFT completion n=4 partial sum after coefficient1 = 426

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

recurrence LEFT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[20, 91];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 20, 60], [1, 3, 4, 91, 364]]

After

new a_4=426;next state=[91, 426];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S016

S018 · calculation

Before

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 91)

After

recurrence LEFT completion n=5 ORIGINAL coefficient0 times a_3 = 273

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

recurrence LEFT completion n=5 partial sum after coefficient0 := sum(2, 273)

After

recurrence LEFT completion n=5 partial sum after coefficient0 = 275

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 426)

After

recurrence LEFT completion n=5 ORIGINAL coefficient1 times a_4 = 1704

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

recurrence LEFT completion n=5 partial sum after coefficient1 := sum(275, 1704)

After

recurrence LEFT completion n=5 partial sum after coefficient1 = 1979

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

recurrence LEFT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[91, 426];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 91, 273], [1, 4, 4, 426, 1704]]

After

new a_5=1979;next state=[426, 1979];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S021

S023 · calculation

Before

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 426)

After

recurrence LEFT completion n=6 ORIGINAL coefficient0 times a_4 = 1278

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

recurrence LEFT completion n=6 partial sum after coefficient0 := sum(2, 1278)

After

recurrence LEFT completion n=6 partial sum after coefficient0 = 1280

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 1979)

After

recurrence LEFT completion n=6 ORIGINAL coefficient1 times a_5 = 7916

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

recurrence LEFT completion n=6 partial sum after coefficient1 := sum(1280, 7916)

After

recurrence LEFT completion n=6 partial sum after coefficient1 = 9196

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

recurrence LEFT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[426, 1979];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 426, 1278], [1, 5, 4, 1979, 7916]]

After

new a_6=9196;next state=[1979, 9196];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S026

S028 · calculation

Before

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 1979)

After

recurrence LEFT completion n=7 ORIGINAL coefficient0 times a_5 = 5937

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

recurrence LEFT completion n=7 partial sum after coefficient0 := sum(2, 5937)

After

recurrence LEFT completion n=7 partial sum after coefficient0 = 5939

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 9196)

After

recurrence LEFT completion n=7 ORIGINAL coefficient1 times a_6 = 36784

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

recurrence LEFT completion n=7 partial sum after coefficient1 := sum(5939, 36784)

After

recurrence LEFT completion n=7 partial sum after coefficient1 = 42723

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · implies

Before

recurrence LEFT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[1979, 9196];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 1979, 5937], [1, 6, 4, 9196, 36784]]

After

new a_7=42723;next state=[9196, 42723];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196, 42723]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S031

S033 · calculation

Before

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 9196)

After

recurrence LEFT completion n=8 ORIGINAL coefficient0 times a_6 = 27588

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

recurrence LEFT completion n=8 partial sum after coefficient0 := sum(2, 27588)

After

recurrence LEFT completion n=8 partial sum after coefficient0 = 27590

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 42723)

After

recurrence LEFT completion n=8 ORIGINAL coefficient1 times a_7 = 170892

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

recurrence LEFT completion n=8 partial sum after coefficient1 := sum(27590, 170892)

After

recurrence LEFT completion n=8 partial sum after coefficient1 = 198482

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · implies

Before

recurrence LEFT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[9196, 42723];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 9196, 27588], [1, 7, 4, 42723, 170892]]

After

new a_8=198482;next state=[42723, 198482];drop ONLY oldest state entry then append new scalar;prefix now=[2, 3, 20, 91, 426, 1979, 9196, 42723, 198482]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S036

S038 · implies

Before

recurrence LEFT completion:complete ORIGINAL initial state=[2, 3];actual generated prefix=[2, 3, 20, 91, 426, 1979, 9196, 42723, 198482]

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 dependencies

Rule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S037

S039 · implies

Before

recurrence RIGHT completion:ORIGINAL order=2;oldest-to-newest coefficients=[3, 4];constant=2;ALL initial a_0..a_(r-1)=[2, 4]

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 dependencies

Rule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S038

S040 · calculation

Before

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 := mul(3, 2)

After

recurrence RIGHT completion n=2 ORIGINAL coefficient0 times a_0 = 6

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

recurrence RIGHT completion n=2 partial sum after coefficient0 := sum(2, 6)

After

recurrence RIGHT completion n=2 partial sum after coefficient0 = 8

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 := mul(4, 4)

After

recurrence RIGHT completion n=2 ORIGINAL coefficient1 times a_1 = 16

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

recurrence RIGHT completion n=2 partial sum after coefficient1 := sum(8, 16)

After

recurrence RIGHT completion n=2 partial sum after coefficient1 = 24

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=2;previous state a_(n-r)..a_(n-1)=[2, 4];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 3, 2, 6], [1, 1, 4, 4, 16]]

After

new a_2=24;next state=[4, 24];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S043

S045 · calculation

Before

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 := mul(3, 4)

After

recurrence RIGHT completion n=3 ORIGINAL coefficient0 times a_1 = 12

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

recurrence RIGHT completion n=3 partial sum after coefficient0 := sum(2, 12)

After

recurrence RIGHT completion n=3 partial sum after coefficient0 = 14

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 := mul(4, 24)

After

recurrence RIGHT completion n=3 ORIGINAL coefficient1 times a_2 = 96

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

recurrence RIGHT completion n=3 partial sum after coefficient1 := sum(14, 96)

After

recurrence RIGHT completion n=3 partial sum after coefficient1 = 110

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=3;previous state a_(n-r)..a_(n-1)=[4, 24];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 3, 4, 12], [1, 2, 4, 24, 96]]

After

new a_3=110;next state=[24, 110];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S048

S050 · calculation

Before

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 := mul(3, 24)

After

recurrence RIGHT completion n=4 ORIGINAL coefficient0 times a_2 = 72

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

recurrence RIGHT completion n=4 partial sum after coefficient0 := sum(2, 72)

After

recurrence RIGHT completion n=4 partial sum after coefficient0 = 74

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 := mul(4, 110)

After

recurrence RIGHT completion n=4 ORIGINAL coefficient1 times a_3 = 440

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

recurrence RIGHT completion n=4 partial sum after coefficient1 := sum(74, 440)

After

recurrence RIGHT completion n=4 partial sum after coefficient1 = 514

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[24, 110];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 3, 24, 72], [1, 3, 4, 110, 440]]

After

new a_4=514;next state=[110, 514];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S053

S055 · calculation

Before

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 := mul(3, 110)

After

recurrence RIGHT completion n=5 ORIGINAL coefficient0 times a_3 = 330

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

recurrence RIGHT completion n=5 partial sum after coefficient0 := sum(2, 330)

After

recurrence RIGHT completion n=5 partial sum after coefficient0 = 332

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · calculation

Before

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 := mul(4, 514)

After

recurrence RIGHT completion n=5 ORIGINAL coefficient1 times a_4 = 2056

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S056

S058 · calculation

Before

recurrence RIGHT completion n=5 partial sum after coefficient1 := sum(332, 2056)

After

recurrence RIGHT completion n=5 partial sum after coefficient1 = 2388

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[110, 514];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 3, 110, 330], [1, 4, 4, 514, 2056]]

After

new a_5=2388;next state=[514, 2388];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S058

S060 · calculation

Before

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 := mul(3, 514)

After

recurrence RIGHT completion n=6 ORIGINAL coefficient0 times a_4 = 1542

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

recurrence RIGHT completion n=6 partial sum after coefficient0 := sum(2, 1542)

After

recurrence RIGHT completion n=6 partial sum after coefficient0 = 1544

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 := mul(4, 2388)

After

recurrence RIGHT completion n=6 ORIGINAL coefficient1 times a_5 = 9552

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

recurrence RIGHT completion n=6 partial sum after coefficient1 := sum(1544, 9552)

After

recurrence RIGHT completion n=6 partial sum after coefficient1 = 11096

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[514, 2388];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 3, 514, 1542], [1, 5, 4, 2388, 9552]]

After

new a_6=11096;next state=[2388, 11096];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S063

S065 · calculation

Before

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 := mul(3, 2388)

After

recurrence RIGHT completion n=7 ORIGINAL coefficient0 times a_5 = 7164

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

recurrence RIGHT completion n=7 partial sum after coefficient0 := sum(2, 7164)

After

recurrence RIGHT completion n=7 partial sum after coefficient0 = 7166

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 := mul(4, 11096)

After

recurrence RIGHT completion n=7 ORIGINAL coefficient1 times a_6 = 44384

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

recurrence RIGHT completion n=7 partial sum after coefficient1 := sum(7166, 44384)

After

recurrence RIGHT completion n=7 partial sum after coefficient1 = 51550

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[2388, 11096];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 3, 2388, 7164], [1, 6, 4, 11096, 44384]]

After

new a_7=51550;next state=[11096, 51550];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096, 51550]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S068

S070 · calculation

Before

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 := mul(3, 11096)

After

recurrence RIGHT completion n=8 ORIGINAL coefficient0 times a_6 = 33288

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

recurrence RIGHT completion n=8 partial sum after coefficient0 := sum(2, 33288)

After

recurrence RIGHT completion n=8 partial sum after coefficient0 = 33290

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 := mul(4, 51550)

After

recurrence RIGHT completion n=8 ORIGINAL coefficient1 times a_7 = 206200

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

recurrence RIGHT completion n=8 partial sum after coefficient1 := sum(33290, 206200)

After

recurrence RIGHT completion n=8 partial sum after coefficient1 = 239490

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · implies

Before

recurrence RIGHT completion:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[11096, 51550];ALL [j,source_index,coefficient,value,product] rows=[[0, 6, 3, 11096, 33288], [1, 7, 4, 51550, 206200]]

After

new a_8=239490;next state=[51550, 239490];drop ONLY oldest state entry then append new scalar;prefix now=[2, 4, 24, 110, 514, 2388, 11096, 51550, 239490]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S073

S075 · implies

Before

recurrence RIGHT completion:complete ORIGINAL initial state=[2, 4];actual generated prefix=[2, 4, 24, 110, 514, 2388, 11096, 51550, 239490]

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

Step identifiers and dependencies

Rule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S074

S076 · implies

Before

ORIGINAL only known a_0=2;different completions a_1=3 and4;target=8

After

WHOLE prefixes including initials differ=Yes;actual targets=[198482, 239490] 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 dependencies

Rule: 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

(2, 3, 4, [2, 3, 20, 91, 426, 1979, 9196, 42723, 198482], [2, 4, 24, 110, 514, 2388, 11096, 51550, 239490], 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: 8a068b2d7c46767d97ffde1446a7ebbef356c1d5917b9b76dcc68c09ad8bc104
candidate record SHA-256: 5d376cb88b4a2a1320106dc6ba7b3aeeae93f4bc1264cb62ae9c5ac61895f537
semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
Lean: NOT_RUN · human review: NOT_PERFORMED

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