Representations of n as a sum of generalized heptagonal, octagonal, and nonagonal numbers

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-255916,
  title        = {Representations of n as a sum of generalized heptagonal, octagonal, and nonagonal numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-255916}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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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 (i): For $k \geq j \geq 3$ , every nonnegative integer can be written as the sum of a generalized heptagonal number, a $j$ -gonal number and a $k$ -gonal number, if and only if $(j, k)$ is among the following ordered pairs: $(3, k)$ ( $k = 3..19, 21..24, 26, 27, 29, 30$ ), $(4, k)$ ( $k = 4..11, 13, 14, 17, 19, 20, 23, 26$ ), $(5, 6), (5, 9), (6, 7), (8, 9)$ .

_Zhi-Wei Sun_

conjecture_3. Conjecture (ii): For $k \geq j \geq 3$ , every nonnegative integer can be written as the sum of a generalized pentagonal number, a $j$ -gonal number and a $k$ -gonal number, if and only if $(j, k)$ is among the following ordered pairs: $(3, k)$ ( $k = 3..20, 22, 24, 25, 28..30, 32, 37$ ), $(4, k)$ ( $k = 4..13, 15, 16, 18, 20..25, 27, 28, 31, 33, 34$ ), $(5, k)$ ( $k = 6..12, 20$ ), $(6, k)$ ( $k = 7..10$ ), $(7, 9), (7, 11), (8, 10), (9, 11)$ .

_Zhi-Wei Sun_

The sequence $a (n)$ counts the number of ways to write $n$ as the sum of a generalized heptagonal number $G_{7} (k) = \frac{5 k ^{2} - 3 k}{2}$ ( $k \in Z$ ), an octagonal number $P_{8} (x) = x (3 x - 2)$ ( $x \geq 0$ ), and a nonagonal number $P_{9} (y) = \frac{y ( 7 y - 5 )}{2}$ ( $y \geq 0$ ).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«255916» (3 statements).

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

theorem conjecture_2 (j k : ℕ) (hjk : 3 ≤ j ∧ j ≤ k) :
    (∀ n : ℕ, IsSumOfGmPjPk 7 n j k) ↔ (j, k) ∈ sunPolygonalPairsSet1

theorem conjecture_3 (j k : ℕ) (hjk : 3 ≤ j ∧ j ≤ k) :
    (∀ n : ℕ, IsSumOfGmPjPk 5 n j k) ↔ (j, k) ∈ sunPolygonalPairsSet2

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).
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

A255916

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.