Representations of n as the sum of two pentagonal and a generalized decagonal number

Conjecture: a(n) > 0 for all n. - _Zhi-Wei Sun_

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.

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Cite

@misc{cairn-oeis-253187,
  title        = {Representations of n as the sum of two pentagonal and a generalized decagonal number},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-253187}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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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_1. Conjecture: $a (n) > 0$ for all $n$ .

_Zhi-Wei Sun_

conjecture_2. Conjecture: For any ordered pair $(k, m)$ among $(5, 7), (5, 9), (5, 13), (6, 5), (6, 7), (7, 5)$ , each nonnegative integer $n$ can be written as the sum of a $k$ -gonal number, a second $k$ -gonal number, and a generalized $m$ -gonal number.

_Zhi-Wei Sun_

The sequence $a (n)$ counts the number of ordered ways to write $n$ as the sum of a pentagonal number $\frac{x ( 3 x - 1 )}{2}$ ( $x \geq 0$ ), a second pentagonal number $\frac{y ( 3 y + 1 )}{2}$ ( $y \geq 0$ ), and a generalized decagonal number $m (4 m - 3)$ ( $m \in Z$ ).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«253187» (2 statements).

theorem conjecture_1 (n : ℕ) : 0 < a n

theorem conjecture_2 (k m : ℕ) (hkm : (k, m) ∈ cPairs) (n : ℕ) :
    ∃ x y : ℕ, ∃ z : ℤ,
      Nat.polygonalNumber k x + Nat.generalizedPolygonalNumber k (-(y : ℤ)) +
        Nat.generalizedPolygonalNumber m z = n

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
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A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A253187

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.