Alphabeta Math
CorollaryStatement: 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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Binary words avoiding any fixed nonempty factor have a rational length generating function

Statement

Fix a nonempty binary word w. If an is the number of words in {0,1}n that do not contain w as a contiguous factor, then ∑n≥0anxn is a rational formal power series over Z.

Facts & Assumptions

Given: A nonempty word w over the finite alphabet {0,1}.

[L1]

Words over a finite alphabet that avoid a finite set of nonempty factors have a rational length generating function (Words over a finite alphabet avoiding finitely many nonempty factors have a rational length generating function).

Proof

technique · specialization
1.1

The alphabet {0,1} is finite, and B={w} is a finite set of nonempty words.

given
2.1

Apply [L1] using step 1.1. Its conclusion is the asserted rationality.

step 1.1L1∎

Depends on

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: 19 results over 6 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