formal-conjectures · oeis · ams-11
The sequence is defined by
$$a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4.$$
Conjecture: Given $F(n,L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$, then
$\lim_{n\to\infty} F(n+1,L)/F(n,L) = (\mathrm{Fibonacci}(L)\sqrt{5} + \mathrm{Lucas}(L))/2$ for
$L \ge 0$ where $\mathrm{Fibonacci}(n) = \mathrm{A000045}(n)$ and
$\mathrm{Lucas}(n) = \mathrm{A000032}(n)$.
Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).
Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.
Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A181546
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/181546.lean
Local target: Corpus.OEIS181546.conjecture
Source SHA-256: 3672405fe4cacc4ad906c2ff7a7e94b8e70cd2b9d6d9a0f4c835da1d493c4634
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.
Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.
Conjecture: Given $F(n,L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$, then $\lim_{n\to\infty} F(n+1,L)/F(n,L) = (\mathrm{Fibonacci}(L)\sqrt{5} + \mathrm{Lucas}(L))/2$ for $L \ge 0$ where $\mathrm{Fibonacci}(n) = \mathrm{A000045}(n)$ and $\mathrm{Lucas}(n) = \mathrm{A000
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