Question 22540: A Complete Initial State Gives a Unique Forward Prefix

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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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R0044-22540 · q-20261014-advance-20261110-r05-recurrence-relation-0065

Original question and givens

Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

G001 — Use order r=4, coefficients=[1, 2, 3, 4] in oldest-to-newest order, constant=2 and initial_values=[14, 15, 16, 17], where initial_values[j]=a_j. For n≥r, a_n=constant+sum over j=0..r−1 of coefficients[j]*a_(n−r+j). The state shifts left then appends the new scalar. All intermediate integer operations must be safe. Target index=9. Give (order, initial_count, target_n, target_value, unique_prefix, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

mko-recurrence-relation

Step-by-step derivation

S001 · implies

Before

source operation=recurrence-initial-value-uniqueness;original snapshot and mathematics retained

After

explicit ordered source output fields=[order, initial_count, target_n, target_value, unique_prefix, certificate]

Conditions: none

Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

recurrence ORIGINAL:ORIGINAL order=4;oldest-to-newest coefficients=[1, 2, 3, 4];constant=2;ALL initial a_0..a_(r-1)=[14, 15, 16, 17]

After

for EVERY n>=4,a_n=constant+sum_j coefficients[j]*a_(n-r+j);current prefix length=4;forward rule is total and single-valued

Conditions: none

Fix all r initials and the oldest-to-newest coefficient roles.Given the preceding r values,the next value is uniquely determined.This is a forward definition,not a closed form or convergence claim.

Step identifiers and dependencies

Rule: recurrence_complete_initial_state_and_total_single_value_rule · Uses: G001, S001

S003 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 := mul(1, 14)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 = 14

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient0 := sum(2, 14)

After

recurrence ORIGINAL n=4 partial sum after coefficient0 = 16

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 := mul(2, 15)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 = 30

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient1 := sum(16, 30)

After

recurrence ORIGINAL n=4 partial sum after coefficient1 = 46

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 := mul(3, 16)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 = 48

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient2 := sum(46, 48)

After

recurrence ORIGINAL n=4 partial sum after coefficient2 = 94

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 := mul(4, 17)

After

recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 = 68

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

recurrence ORIGINAL n=4 partial sum after coefficient3 := sum(94, 68)

After

recurrence ORIGINAL n=4 partial sum after coefficient3 = 162

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=4;previous state a_(n-r)..a_(n-1)=[14, 15, 16, 17];ALL [j,source_index,coefficient,value,product] rows=[[0, 0, 1, 14, 14], [1, 1, 2, 15, 30], [2, 2, 3, 16, 48], [3, 3, 4, 17, 68]]

After

new a_4=162;next state=[15, 16, 17, 162];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162]

Conditions: none

Multiply each coefficient by its specified original index,add to the constant and check the source safe-integer range.Shift left and append the new scalar;do not reverse coefficients or feed the appended value back into this same update.

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S010

S012 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 := mul(1, 15)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 = 15

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient0 := sum(2, 15)

After

recurrence ORIGINAL n=5 partial sum after coefficient0 = 17

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 := mul(2, 16)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 = 32

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient1 := sum(17, 32)

After

recurrence ORIGINAL n=5 partial sum after coefficient1 = 49

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 := mul(3, 17)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 = 51

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient2 := sum(49, 51)

After

recurrence ORIGINAL n=5 partial sum after coefficient2 = 100

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 := mul(4, 162)

After

recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 = 648

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

recurrence ORIGINAL n=5 partial sum after coefficient3 := sum(100, 648)

After

recurrence ORIGINAL n=5 partial sum after coefficient3 = 748

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=5;previous state a_(n-r)..a_(n-1)=[15, 16, 17, 162];ALL [j,source_index,coefficient,value,product] rows=[[0, 1, 1, 15, 15], [1, 2, 2, 16, 32], [2, 3, 3, 17, 51], [3, 4, 4, 162, 648]]

After

new a_5=748;next state=[16, 17, 162, 748];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S019

S021 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 := mul(1, 16)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 = 16

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient0 := sum(2, 16)

After

recurrence ORIGINAL n=6 partial sum after coefficient0 = 18

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 := mul(2, 17)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 = 34

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient1 := sum(18, 34)

After

recurrence ORIGINAL n=6 partial sum after coefficient1 = 52

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 := mul(3, 162)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 = 486

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient2 := sum(52, 486)

After

recurrence ORIGINAL n=6 partial sum after coefficient2 = 538

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 := mul(4, 748)

After

recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 = 2992

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

recurrence ORIGINAL n=6 partial sum after coefficient3 := sum(538, 2992)

After

recurrence ORIGINAL n=6 partial sum after coefficient3 = 3530

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=6;previous state a_(n-r)..a_(n-1)=[16, 17, 162, 748];ALL [j,source_index,coefficient,value,product] rows=[[0, 2, 1, 16, 16], [1, 3, 2, 17, 34], [2, 4, 3, 162, 486], [3, 5, 4, 748, 2992]]

After

new a_6=3530;next state=[17, 162, 748, 3530];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S028

S030 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 := mul(1, 17)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 = 17

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient0 := sum(2, 17)

After

recurrence ORIGINAL n=7 partial sum after coefficient0 = 19

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 := mul(2, 162)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 = 324

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient1 := sum(19, 324)

After

recurrence ORIGINAL n=7 partial sum after coefficient1 = 343

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 := mul(3, 748)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 = 2244

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient2 := sum(343, 2244)

After

recurrence ORIGINAL n=7 partial sum after coefficient2 = 2587

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 := mul(4, 3530)

After

recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 = 14120

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

recurrence ORIGINAL n=7 partial sum after coefficient3 := sum(2587, 14120)

After

recurrence ORIGINAL n=7 partial sum after coefficient3 = 16707

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=7;previous state a_(n-r)..a_(n-1)=[17, 162, 748, 3530];ALL [j,source_index,coefficient,value,product] rows=[[0, 3, 1, 17, 17], [1, 4, 2, 162, 324], [2, 5, 3, 748, 2244], [3, 6, 4, 3530, 14120]]

After

new a_7=16707;next state=[162, 748, 3530, 16707];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S037

S039 · calculation

Before

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 := mul(1, 162)

After

recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 = 162

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

recurrence ORIGINAL n=8 partial sum after coefficient0 := sum(2, 162)

After

recurrence ORIGINAL n=8 partial sum after coefficient0 = 164

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 := mul(2, 748)

After

recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 = 1496

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

recurrence ORIGINAL n=8 partial sum after coefficient1 := sum(164, 1496)

After

recurrence ORIGINAL n=8 partial sum after coefficient1 = 1660

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 := mul(3, 3530)

After

recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 = 10590

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

recurrence ORIGINAL n=8 partial sum after coefficient2 := sum(1660, 10590)

After

recurrence ORIGINAL n=8 partial sum after coefficient2 = 12250

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 := mul(4, 16707)

After

recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 = 66828

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

recurrence ORIGINAL n=8 partial sum after coefficient3 := sum(12250, 66828)

After

recurrence ORIGINAL n=8 partial sum after coefficient3 = 79078

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=8;previous state a_(n-r)..a_(n-1)=[162, 748, 3530, 16707];ALL [j,source_index,coefficient,value,product] rows=[[0, 4, 1, 162, 162], [1, 5, 2, 748, 1496], [2, 6, 3, 3530, 10590], [3, 7, 4, 16707, 66828]]

After

new a_8=79078;next state=[748, 3530, 16707, 79078];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S046

S048 · calculation

Before

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 := mul(1, 748)

After

recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 = 748

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

recurrence ORIGINAL n=9 partial sum after coefficient0 := sum(2, 748)

After

recurrence ORIGINAL n=9 partial sum after coefficient0 = 750

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 := mul(2, 3530)

After

recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 = 7060

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

recurrence ORIGINAL n=9 partial sum after coefficient1 := sum(750, 7060)

After

recurrence ORIGINAL n=9 partial sum after coefficient1 = 7810

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 := mul(3, 16707)

After

recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 = 50121

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

recurrence ORIGINAL n=9 partial sum after coefficient2 := sum(7810, 50121)

After

recurrence ORIGINAL n=9 partial sum after coefficient2 = 57931

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 := mul(4, 79078)

After

recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 = 316312

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

recurrence ORIGINAL n=9 partial sum after coefficient3 := sum(57931, 316312)

After

recurrence ORIGINAL n=9 partial sum after coefficient3 = 374243

Conditions: none

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

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · implies

Before

recurrence ORIGINAL:ORIGINAL target n=9;previous state a_(n-r)..a_(n-1)=[748, 3530, 16707, 79078];ALL [j,source_index,coefficient,value,product] rows=[[0, 5, 1, 748, 748], [1, 6, 2, 3530, 7060], [2, 7, 3, 16707, 50121], [3, 8, 4, 79078, 316312]]

After

new a_9=374243;next state=[3530, 16707, 79078, 374243];drop ONLY oldest state entry then append new scalar;prefix now=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243]

Conditions: none

Step identifiers and dependencies

Rule: recurrence_EVERY_state_update_shift_left_append_exact · Uses: G001, S055

S057 · implies

Before

recurrence ORIGINAL:complete ORIGINAL initial state=[14, 15, 16, 17];actual generated prefix=[14, 15, 16, 17, 162, 748, 3530, 16707, 79078, 374243]

After

ANY two sequences with SAME r initials and SAME total rule agree at n=4 by substitution,then at every later n by induction;this determines the requested prefix only

Conditions: none

Induct from equality of all r initials:the same single-valued rule gives equality of the next term,and hence every later prefix term.Finite generation proves no closed form,asymptotic behavior or analyticity.

Step identifiers and dependencies

Rule: recurrence_forward_uniqueness_induction_not_analytic_claim · Uses: G001, S056

Reverse checks

S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S056 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S055 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S054 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient3 times a_8 with exact rational arithmetic, not rounded decimals.
S053 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S052 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient2 times a_7 with exact rational arithmetic, not rounded decimals.
S051 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S050 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient1 times a_6 with exact rational arithmetic, not rounded decimals.
S049 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S048 · premise_and_result_audit — Recompute recurrence ORIGINAL n=9 ORIGINAL coefficient0 times a_5 with exact rational arithmetic, not rounded decimals.
S047 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S046 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S045 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient3 times a_7 with exact rational arithmetic, not rounded decimals.
S044 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S043 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient2 times a_6 with exact rational arithmetic, not rounded decimals.
S042 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S041 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient1 times a_5 with exact rational arithmetic, not rounded decimals.
S040 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S039 · premise_and_result_audit — Recompute recurrence ORIGINAL n=8 ORIGINAL coefficient0 times a_4 with exact rational arithmetic, not rounded decimals.
S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S037 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S036 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient3 times a_6 with exact rational arithmetic, not rounded decimals.
S035 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S034 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient2 times a_5 with exact rational arithmetic, not rounded decimals.
S033 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S032 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient1 times a_4 with exact rational arithmetic, not rounded decimals.
S031 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S030 · premise_and_result_audit — Recompute recurrence ORIGINAL n=7 ORIGINAL coefficient0 times a_3 with exact rational arithmetic, not rounded decimals.
S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S028 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S027 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient3 times a_5 with exact rational arithmetic, not rounded decimals.
S026 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S025 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient2 times a_4 with exact rational arithmetic, not rounded decimals.
S024 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S023 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient1 times a_3 with exact rational arithmetic, not rounded decimals.
S022 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S021 · premise_and_result_audit — Recompute recurrence ORIGINAL n=6 ORIGINAL coefficient0 times a_2 with exact rational arithmetic, not rounded decimals.
S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S019 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S018 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient3 times a_4 with exact rational arithmetic, not rounded decimals.
S017 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S016 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient2 times a_3 with exact rational arithmetic, not rounded decimals.
S015 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S014 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient1 times a_2 with exact rational arithmetic, not rounded decimals.
S013 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S012 · premise_and_result_audit — Recompute recurrence ORIGINAL n=5 ORIGINAL coefficient0 times a_1 with exact rational arithmetic, not rounded decimals.
S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S010 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient3 with exact rational arithmetic, not rounded decimals.
S009 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient3 times a_3 with exact rational arithmetic, not rounded decimals.
S008 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient2 with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient2 times a_2 with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient1 with exact rational arithmetic, not rounded decimals.
S005 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient1 times a_1 with exact rational arithmetic, not rounded decimals.
S004 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 partial sum after coefficient0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Recompute recurrence ORIGINAL n=4 ORIGINAL coefficient0 times a_0 with exact rational arithmetic, not rounded decimals.
S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(4, 4, 9, 374243, Yes, total_single_valued_rule_plus_all_order_many_initial_values_gives_unique_prefix)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts

input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
original question SHA-256: dbc4b131fb556e5fb0f90934f539554b8ca1fa45bf989dfb79956217e82404f9
candidate record SHA-256: 77fe653e125038a84c9c29ef4c786ca331576aea1eb2526f1bca2f50b27f7e61
semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
Lean: NOT_RUN · human review: NOT_PERFORMED

Same-source full JSON