Question 24696: The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF

Original question and givens

Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=21, a_1=22, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).

• G001 — Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=21, a_1=22, and x=1/6 with |x|<1/2. Give (a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate).

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-analytic-ordinary-generating-function

Step-by-step derivation

source roles = [ogf-recurrence-boundary-terms]

derived = [[a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate]]

OGF complete recurrence coefficient2 := sum(21, 22)

OGF complete recurrence coefficient2 = 43

OGF complete recurrence coefficient3 := sum(22, 43)

OGF complete recurrence coefficient3 = 65

OGF complete recurrence coefficient4 := sum(43, 65)

OGF complete recurrence coefficient4 = 108

OGF complete recurrence coefficient5 := sum(65, 108)

OGF complete recurrence coefficient5 = 173

OGF complete recurrence coefficient6 := sum(108, 173)

OGF complete recurrence coefficient6 = 281

OGF complete recurrence coefficient7 := sum(173, 281)

OGF complete recurrence coefficient7 = 454

OGF complete recurrence coefficient8 := sum(281, 454)

OGF complete recurrence coefficient8 = 735

OGF complete recurrence coefficient9 := sum(454, 735)

OGF complete recurrence coefficient9 = 1189

OGF initial a1 minus a0 boundary := sub(22, 21)

OGF initial a1 minus a0 boundary = 1

OGF initial boundary times x := mul(1, 1/6)

OGF initial boundary times x = 1/6

OGF numerator with BOTH initial terms := sum(21, 1/6)

OGF numerator with BOTH initial terms = 127/6

(1/6)^2

OGF original x square = 1/36

OGF denominator one minus x := sub(1, 1/6)

OGF denominator one minus x = 5/6

OGF denominator subtract x squared := sub(5/6, 1/36)

OGF denominator subtract x squared = 29/36

OGF recurrence analytic closed value := div(127/6, 29/36)

OGF recurrence analytic closed value = 762/29

source roles = [21, 22, 1/6, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189]]

derived = [127/6, 29/36, 762/29, (1-x-x^2)A=a0+(a1-a0)x]

Reverse checks

• 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 OGF recurrence analytic closed value with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute OGF denominator subtract x squared with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute OGF denominator one minus x with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute OGF numerator with BOTH initial terms with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute OGF initial boundary times x with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute OGF initial a1 minus a0 boundary with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute OGF complete recurrence coefficient9 with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute OGF complete recurrence coefficient8 with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute OGF complete recurrence coefficient7 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute OGF complete recurrence coefficient6 with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute OGF complete recurrence coefficient5 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute OGF complete recurrence coefficient4 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute OGF complete recurrence coefficient3 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute OGF complete recurrence coefficient2 with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(21, 22, [21, 22, 43, 65, 108, 173, 281, 454, 735, 1189], 1/6, 127/6, 29/36, 762/29, A(x)=(a0+(a1-a0)x)/(1-x-x^2)_with_initial_boundary_terms_preserved)

The 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.