PropositionStatement: Literature-sourcedProof: 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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Changing finitely many coefficients preserves rationality and eventual linear recurrence

Statement

Let $(a_{n})$ and $(b_{n})$ be sequences over a field which differ at only finitely many indices. Then the generating series $\sum a_{n} x^{n}$ is rational if and only if $\sum b_{n} x^{n}$ is rational. Equivalently, $(a_{n})$ is eventually linearly recurrent if and only if $(b_{n})$ is eventually linearly recurrent.

A finite modification need not preserve a recurrence with starting index zero.

Facts & Assumptions

Given: Two sequences $(a_{n})$ and $(b_{n})$ over a field which differ at only finitely many indices.

[L1]

A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational (A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational).

Proof

technique · direct

1.1

The difference $H (x) = \sum_{n \geq 0} (a_{n} - b_{n}) x^{n}$ has finite support, so it is a polynomial.

given

2.1

If $A (x) = \sum a_{n} x^{n} = P / Q$ is rational, then $B (x) = A (x) - H (x) = (P - H Q) / Q$ is rational; the same argument with $A = B + H$ proves the converse.

step 1.1algebra

3.1

Applying [L1] to both series converts step 2.1 into the equivalence of eventual recurrence.

step 2.1L1

4.1

For the final warning, the zero sequence satisfies $a_{n + 1} + a_{n} = 0$ from zero, while changing only $a_{0}$ to $1$ destroys that identity at $n = 0$ , even though the modified sequence is eventually zero.

step 3.1∎

Depends on

A coefficient sequence is eventually linearly recurrent if and only if its formal generating function is rational

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: 11 results over 4 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., Proposition 4.2.2 (standard reference, not scraped)
M. Waldschmidt, Linear Recurrence Sequences VI, ultimately recurrent sequences (standard reference, not scraped)