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Metadata 
bibkey: schulte2025a385590
authors: Werner Schulte
year: 2025
title: OEIS A385590, a triangle based on Fibonacci numbers
doi: null
url: https://oeis.org/A385590
claim: The alternating binomial sum of row n is zero for n greater than two and is (-1)^n for n less than three.
strata_touched:
  - D5/S1/Recurrence/Parity/A385590
license: citation-only
triage: anchor

OEIS A385590

The entry was authored on 2025-07-03. On 2026-09-08 the worker retrieved the internal-format page and read its definition and formula field directly. The formula field still says:

Conjecture: Sum_{k=1..n} (-1)^k * binomial(n-1, k-1) * T(n, k) = 0 for n > 2 and (-1)^n for n < 3.

The row index is the unique i > 1 with F(i) <= n < F(i+1), and T(n,k) = F(i-1)^2 + 1 - ((i-1) mod 2) + (n-F(i))*F(i-2)

  • (k-1)*F(i-1). Here F(0)=0 and F(1)=1. Rows begin at n=1. The separate comment that the triangle permutes the natural numbers is not the statement addressed by this module.

The statement and triangle definition come from this entry. The proof is derived in this repository by cancelling the constant and linear moments of the alternating binomial coefficients. This note does not attribute that proof to Schulte or OEIS.

Search log

  • Worker, 2026-09-08: HTTPS retrieval of the internal page succeeded; the definition, conjectured sum, author, and date agree with the brief.
  • The caller reports searches by entry number, formula, and Fibonacci triangle keywords in public indexes, OEIS, arXiv and journal pages, with no independent proof found. This is caller-supplied search evidence.
  • Worker: GitHub code search for 385590 language:Lean required sign-in; Google returned an access fallback; Bing returned unrelated results. These requests are not evidence of an exhaustive negative search.
  • No DOI is supplied by the entry. None is invented here. The locator below is the source identity.

Verified locator

  • Entry: https://oeis.org/A385590
  • Definition and formula field: https://oeis.org/A385590/internal (retrieved directly on 2026-09-08).

The source-to-Lean correspondence was checked by reading; the Lean kernel checks the formal statement, not the correspondence with external prose.