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bibkey: kagey2025a382814 authors: Peter Kagey year: 2025 title: “OEIS A382814: first-player score in the Fibonachos game” doi: null url: https://oeis.org/A382814 claim: “a(n) = n/2 exactly at 2,8,10,32; for n>32 the first player wins exactly in odd Fibonacci blocks.” strata_touched:

  • D5/S3/Arith/FibonacciAtomic/FibonachosScore license: citation-only triage: anchor

OEIS A382814

The entry is signed Peter Kagey, 5 April 2025. Its comment block states:

Conjecture: a(n) = n/2 if and only if n is in {2, 8, 10, 32}.

Conjecture: For n > 32, a(n) > n/2 if and only if F(m)-1 <= n <= F(m+1)-2 for some odd integer m, where F(n) = A000045(n).

The Mathematica program initializes m=n, i=1, p=True, c=0. While the heap is positive it resets i to one when Fibonacci[i]>m, adds the move to c when p is true, subtracts the move from m, increments i, and negates p. The sequence is the returned c. The recursive score pair uses the same move and reset, expressing the player flag by swapping the two score components after each move.

Peter Kagey’s Fibonachos blog post, dated 2025-05-26, discusses the restart-count sequences A280521 and A280523 and the first-player score sequence A382814. Its score conjecture for heaps greater than 32 says that the player taking the last move before the first reset receives the larger final total.

Nathan Fox’s Fibonachos proof, dated 2017-01-28, concerns restart counts. Theorem 1 states that the smallest initial heap requiring n rounds is F(2n+1)-n for n>=1; the initial round is counted, so this means n-1 restarts. Theorem 2 identifies the minimum-heap sequence by A280523(n)=A215004(2n-2). These are statements about rounds and minimum heap sizes, distinct from the two A382814 conjectures about ties and first-player majority.

The internal entry, history page and synthesized b-file were retrieved on 2026-10-03. The internal entry still labels both statements Conjecture and lists a(1),...,a(10)=1,1,2,3,3,4,3,4,4,5, with the individual plays. The text-search interface returned HTTP 403. The history page was accessible with a browser User-Agent; its individual revisions were not exhaustively inspected.

Verified locator

  • https://oeis.org/A382814 : named sequence and comment block.
  • https://oeis.org/A382814/internal : original comments, examples, author and Mathematica program; retrieved with HTTP 200 on 2026-10-03.
  • https://oeis.org/history?seq=A382814 : history page retrieved with HTTP 200 on 2026-10-03; no exhaustive revision inspection is claimed.
  • https://oeis.org/A382814/b382814.txt : synthesized first 70 terms; retrieved with HTTP 200 on 2026-10-03.
  • https://peterkagey.com/blog/2025/05/fibonachos/ : Peter Kagey’s Fibonachos post, dated 2025-05-26; retrieved with HTTP 200 on 2026-10-03.
  • https://oeis.org/A280523/a280523.pdf : Nathan Fox’s Fibonachos proof, dated 2017-01-28; both theorem statements retrieved with HTTP 200 on 2026-10-03.

Issue #12461 records the bounded prior-work check before the formal probe. The additional implementation search found no declarations named Fibonachos or A382814 through Loogle and no Lean code-search matches for those terms on GitHub. These checks do not establish global novelty or publication priority.