Number of ways to express $n$ as sum of square, pentagonal, and hexagonal numbers

formal-conjectures · oeis · ams-11

$$a(n) = |\{(x, y, z) \in \mathbb{N}^3 : x^2 + p_5(y) + p_6(z) = n\}|$$

In April 2009, _Zhi-Wei Sun_ conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \dots$.

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/A160324
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/160324.lean
Local target: Corpus.OEIS160324.conjecture1
Source SHA-256: 869d059ef28744fc3f914005f414d3da1e177a5d6b33531e3bc25d0e378417a9
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