bibkey: oeis2025a110037
authors: OEIS Foundation Inc.; Alan Michael Gómez Calderón
year: 2025
title: A110037 signed non-squashing partition parity and its paperfold difference conjecture
doi: null
url: https://oeis.org/A110037
claim: For n at least two, the signed parity of the distinct non-squashing partition count equals A073089(n) minus A073089(n+1).
strata_touched: []
license: citation-only
triage: anchor

A110037 and the independent paperfold recurrence

Verified locator

url: https://oeis.org/A110037

The formula field of https://oeis.org/A110037/internal credits the difference conjecture to Alan Michael Gómez Calderón, August 19, 2025, with domain n>=2. Its comment identifies the signed count as (-1)^floor(n/2) times A090678(n), the parity of A088567(n). The difference formula is labelled a conjecture.

The entry https://oeis.org/A073089/internal has offset one and recurrence a(1)=0; a(4n+2)=a(8n+7)=a(16n+13)=0; a(4n)=a(8n+3)=a(16n+5)=1; and a(8n+1)=a(4n+1). The positive-index convention restricts a(4n)=1 and the halving branch to n>0. The Lean value at zero is a totalization and does not enter the theorem.

Proof scope

Sloane and Sellers’ Theorem 2 and Corollary 4 supply the count recurrence and parity facts; their separate note is sloane2003nonsquashing. The count is a finite set of finite sets of positive natural parts, with each part at least the sum of smaller parts. Equality is allowed. The paperfold sequence has the independent branch recurrence above.

The proof establishes the maximum-part bijection, the recurrence, and the parity inputs, then applies the strong-induction bridge. No global absence or first-publication priority is claimed. Other OEIS links and other formulas are ASSUMED-UNVERIFIED unless separately cited as read.