False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: The recurrence coefficients alone determine the sequence

Statement

False claim. The coefficients of a constant-coefficient linear recurrence determine its sequence without any initial values.

Facts & Assumptions

Given: The recurrence $a_{n + 2} = a_{n + 1} + a_{n}$ .

[L1]

For a fixed order- $d$ recurrence, every choice of $d$ initial values determines exactly one recurrence sequence ( $d$ initial values determine a sequence satisfying a fixed order- $d$ recurrence).

Refutation

technique · two initial-value witnesses

1.1

With initial pair $(a_{0}, a_{1}) = (0, 0)$ , the recurrence gives the zero sequence. With initial pair $(b_{0}, b_{1}) = (0, 1)$ , it gives $0, 1, 1, 2, 3, \dots$ .

givenalgebra

2.1

The two sequences obey exactly the same recurrence coefficients but differ at index one. This is consistent with [L1], which requires the two initial values in addition to the coefficients, and it refutes the claim.

step 1.1L1∎

Depends on

$d$ initial values determine a sequence satisfying a fixed order-$d$ recurrence

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 6 results over 3 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Sections 4.1-4.2 (standard reference, not scraped)