Normalized convolution of C(6k, 3k)C(3k, k)

Conjecture (ii): For any positive integer n, we have a(n) ≡ 0 pmod 8, and a(n)/8 is odd if and only if n is a power of two. - _Zhi-Wei Sun_, Apr 19 2011

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-189286,
  title        = {Normalized convolution of C(6k, 3k)C(3k, k)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-189286}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Conjecture (ii): For any positive integer $n$ , we have $a (n) \equiv 0 (mod 8)$ , and $a (n) /8$ is odd if and only if $n$ is a power of two.

_Zhi-Wei Sun_, Apr 19 2011

The sequence $a (n)$ is defined by $a (n) = \frac{\sum _{k = 0}^{n} ( 3 k 6 k ) ( k 3 k ) ( 3 ( n - k ) 6 ( n - k ) ) ( n - k 3 ( n - k ) )}{( 2 n - 1 ) ( n 3 n )} .$

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«189286».

theorem conjecture_3 (n : ℕ) (hn : 0 < n) :
    ∃ m : ℤ, a n = 8 * m ∧ (Odd m ↔ ∃ k : ℕ, n = 2 ^ k)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A189286
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.