bibkey: oeis2026a397434 authors: Tanguy Gautier Loic Le Mer year: 2026 title: OEIS A397434, monomial counts for threshold Boolean functions doi: null url: https://oeis.org/A397434 claim: The entry conjectures that adjacent threshold-ANF monomial counts agree exactly when the lower index is two modulo four. strata_touched:

• D5/S3/ArithSums/A397434ThresholdAnf license: citation-only triage: anchor

The adjacent-equality conjecture in A397434

The entry defines a(n) as the number of monomials in the algebraic normal form over GF(2) of the Boolean function that is one exactly when at least ceil(n/2) of its n variables are one. Thus a tie at an even number of variables belongs to the value-one region.

Its formula section labels as a conjecture that a(n) = a(n+1) if and only if n is two modulo four, with a finite verification through index 998. The entry attributes the sequence to Tanguy Gautier Loic Le Mer. Its %F line states exactly: a(2^k+1) = 2^(2^k) - 1 for k >= 1. That separate special-index identity is not a proof of the adjacent-equality classification.

The repository theorem proves the adjacent-equality statement for every natural n >= 1. It makes no assertion of publication priority. Indexed third-party source coverage outside the sources actually opened for this formalization remains ASSUMED-UNVERIFIED.

Locator

• URL: https://oeis.org/A397434