slug: oeis-a385590-alternating-binomial bibkey: schulte2025a385590 doi: null url: https://oeis.org/A385590 triage: theorem motivation_gids:
- D5/S1/Recurrence/FibAlternatingSum
Problem
The single statement considered here is the formula-field conjecture in Werner Schulte’s OEIS A385590, dated 2025-07-03. For every integer n >= 1, let i > 1 be the unique index with F(i) <= n < F(i+1), with F(0)=0 and F(1)=1. Put T(n,k) = F(i-1)^2 + 1 - ((i-1) mod 2) + (n-F(i))*F(i-2)
- (k-1)*F(i-1). Is the sum from k=1 to n of (-1)^k * binomial(n-1,k-1) * T(n,k) equal to (-1)^n when n<3, and zero otherwise? All triangle values and the sum are integers. The entry’s separate permutation conjecture is outside this question.
Motivation
D5/S1/Recurrence/FibAlternatingSum establishes a parity-descending Fibonacci sum. This question concerns a different alternating operation on a triangle whose rows are selected by Fibonacci intervals.
Gap
The parity-descending sum does not give the binomial transform of this triangle. The OEIS formula field still labels this identity Conjecture on 2026-09-08.
Route
Fix the row, write T(n,k)=A+(k-1)B, and cancel the constant and linear moments of the alternating binomial coefficients for n>=3. Evaluate the two boundary rows separately. The Fibonacci inverse must agree with the source’s unique interval index.
Falsifier
A positive n with a different exact integer sum would contradict this statement. A counterexample to the permutation conjecture would not.
Evidence
The caller evaluated n=1 through 12 and obtained -1, 1, and then zeros. These finite observations motivate the question and do not prove it. The worker retrieved https://oeis.org/A385590/internal on 2026-09-08 and checked the formula, author, date, and triangle definition directly.
Triage
Theorem: the required statement quantifies over every positive row, and the affine binomial transform offers a symbolic proof route.
ASSUMED-UNVERIFIED
The source-to-formal-statement correspondence is checked by reading. Public-index searches have not established the absence of every earlier proof; the search limitations are recorded in the linked Library note. No DOI is supplied by the OEIS entry; its stable entry URL is used.