formal-conjectures · oeis · ams-5 · ams-11
The $n$-th term $a(n)$ is given by
$$a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$$
Conjecture: for $n > 0$, $a(n)$ is also the number of sequences of length $n - 1$ covering an
initial interval of positive integers and avoiding three terms
$(\dots, x, \dots, y, \dots, z, \dots)$ such that $x \le y \le z$.
- Gus Wiseman, Jun 17 2021
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/A052709
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/52709.lean
Local target: Corpus.OEIS52709.conjecture
Source SHA-256: 34d894005b2ba8a88c1440b749370694e5fbfff4e066585fed73566232dbf406
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: for $n > 0$, $a(n)$ is also the number of sequences of length $n - 1$ covering an initial interval of positive integers and avoiding three terms $(\dots, x, \dots, y, \dots, z, \dots)$ such that $x \le y \le z$.
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