Number of Abelian cubes of length $3n$ over an alphabet of size 3

formal-conjectures · oeis · ams-11

An Abelian cube is a string of the form $x x' x''$ with $|x| = |x'| = |x''|$ and $x$ is a
permutation of $x'$ and $x''$. The number of Abelian cubes of length $3n$ over an alphabet of
size 3 is given by
$$a(n) = \sum_{k=0}^n \binom{n}{k}^3 \sum_{j=0}^k \binom{k}{j}^3.$$

Conjecture: the supercongruences $a(n \cdot p^k) \equiv a(n \cdot p^{k-1}) \pmod{p^{3k}}$ hold
for primes $p \ge 5$ and positive integers $n$ and $k$.

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/A141057
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/141057.lean
Local target: Corpus.OEIS141057.conjecture1
Source SHA-256: b58619412e1a9d882e69a5460b58e765be085f9ebbf1a84e075f22eda92fa32c
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.

Public JSON record