# Question 24697: The Full Recurrence Keeps Both Initial Boundary Terms in Its OGF [Return to catalogue page 124](https://ocme.evemisslab.com/en/questions/page/124/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261015-advance-20261113-r03-analytic-ordinary-generating-function-foundations/#q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0097) 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. R0048-24697 · q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0097 ## Original question and givens Use the whole recurrence a_n=a_(n−1)+a_(n−2) for every n≥2, initial a_0=22, a_1=23, and x=1/4 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=22, a_1=23, and x=1/4 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](https://ocme.evemisslab.com/en/mko/mko-analytic-ordinary-generating-function/) ## Step-by-step derivation S001 · impliesBefore source roles = [ogf-recurrence-boundary-terms] After derived = [[a0, a1, coefficient_preview, x, numerator, denominator, analytic_value, certificate]] Conditions: none List the full source field order without JavaScript,keeping original data,Boolean flags,mathematical objects and nonclaims distinct. Step identifiers and dependenciesRule: source_ordered_fields_without_JS · Uses: G001 S002 · calculationBefore OGF complete recurrence coefficient2 := sum(22, 23) After OGF complete recurrence coefficient2 = 45 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore OGF complete recurrence coefficient3 := sum(23, 45) After OGF complete recurrence coefficient3 = 68 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore OGF complete recurrence coefficient4 := sum(45, 68) After OGF complete recurrence coefficient4 = 113 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore OGF complete recurrence coefficient5 := sum(68, 113) After OGF complete recurrence coefficient5 = 181 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore OGF complete recurrence coefficient6 := sum(113, 181) After OGF complete recurrence coefficient6 = 294 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore OGF complete recurrence coefficient7 := sum(181, 294) After OGF complete recurrence coefficient7 = 475 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore OGF complete recurrence coefficient8 := sum(294, 475) After OGF complete recurrence coefficient8 = 769 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore OGF complete recurrence coefficient9 := sum(475, 769) After OGF complete recurrence coefficient9 = 1244 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore OGF initial a1 minus a0 boundary := sub(23, 22) After OGF initial a1 minus a0 boundary = 1 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore OGF initial boundary times x := mul(1, 1/4) After OGF initial boundary times x = 1/4 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore OGF numerator with BOTH initial terms := sum(22, 1/4) After OGF numerator with BOTH initial terms = 89/4 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · impliesBefore (1/4)^2 After OGF original x square = 1/16 Conditions: none Evaluate the finite nonnegative-integer power exactly; exponent0 uses the polynomial multiplicative identity. Step identifiers and dependenciesRule: integer_power · Uses: G001, S012 S014 · calculationBefore OGF denominator one minus x := sub(1, 1/4) After OGF denominator one minus x = 3/4 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore OGF denominator subtract x squared := sub(3/4, 1/16) After OGF denominator subtract x squared = 11/16 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore OGF recurrence analytic closed value := div(89/4, 11/16) After OGF recurrence analytic closed value = 356/11 Conditions: none Substitute original roles and established intermediates exactly,checking nonzero denominators without floats. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore source roles = [22, 23, 1/4, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244]] After derived = [89/4, 11/16, 356/11, (1-x-x^2)A=a0+(a1-a0)x] Conditions: none The ten terms preview a whole recurrence.Take C=max(|a0|,|a1|);induction gives |a_n|<=C*2^n,so |x|<1/2 ensures absolute convergence and valid reindexing.Degree0 leaves a0,degree1 leaves(a1−a0)x,and all later coefficients cancel.The denominator exceeds1/4.This sufficient domain is not an exact radius or a termwise-derivative claim. Step identifiers and dependenciesRule: OGF_ALL_n_recurrence_convergence_and_reindex_boundary_terms · Uses: G001, S016 ## 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 (22, 23, [22, 23, 45, 68, 113, 181, 294, 475, 769, 1244], 1/4, 89/4, 11/16, 356/11, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: bd20733412d47921cb773f75c60edde3a26a760859c48aec106c806d1cad61b8 - candidate record SHA-256: b17ffcb5f14392d151c60660fd03f1c281fb4afecc0460ff0b193285ab643b78 - semantic review receipt SHA-256: 049a9e07e69fead88d93b34f0937b9470a06d71c1a8d097058fc83554529bdd0 - question-first witness SHA-256: 0c71ebb418f3de347d232d159fc25e93036f4ab9166bcfb5ee5b4be86d5c55b9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261015-advance-20261113-r03-analytic-ordinary-generating-function-0097.json)