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

The question

conjecture1. In April 2009, _Zhi-Wei Sun_ conjectured that $a(n)>0$ for every $n=0,1,2,3,\dots$.

conjecture2. For each integer $m>2$, any natural number $n$ can be written in the form $p_{m+1}(x_1)+\cdots +p_{2m}(x_m)$ with $x_1,\dots ,x_m$ nonnegative integers, where $p_k(x)=(k-2)x(x-1)/2+x$ ($x=0,1,2,\dots$) are $k$-gonal numbers.

• _Zhi-Wei Sun_, Aug 15 2009

conjecture3. The sequence contains every positive integer.

• _Zhi-Wei Sun_, Sep 04 2009

conjecture4. Conjecture (Zhi-Wei Sun, Aug 21 2009): For any integer $m>2$, each natural number $n$ can be expressed as $p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r$ with $x_1,x_2,x_3\in \mathbb{N}$ and $r\in \{0,\dots ,m-3\}$.

conjecture5. Conjecture (Zhi-Wei Sun, Aug 21 2009): For each integer $m>2$, all sufficiently large integers $n$ can be expressed in the form $p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)$ with $x_1,x_2,x_3\in \mathbb{N}$.

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«160324» (5 statements).

theorem conjecture1 (n : ℕ) : 0 < a n

theorem conjecture2 (m : ℕ) (hm : m > 2) (n : ℕ) :
    ∃ x : Fin m → ℕ, n = ∑ i : Fin m, polygonalNumber (m + (i : ℕ) + 1) (x i)

theorem conjecture3 (k : ℕ) (hk : 0 < k) : ∃ n : ℕ, a n = k

theorem conjecture4 (m : ℕ) (hm : 2 < m) (n : ℕ) :
    ∃ x1 x2 x3 r : ℕ, r ≤ m - 3 ∧
      n = polygonalNumber (m + 1) x1 + polygonalNumber (m + 2) x2 + polygonalNumber (m + 3) x3 + r

theorem conjecture5 (m : ℕ) (hm : 2 < m) :
    ∀ᶠ n in Filter.atTop,
      ∃ x1 x2 x3 : ℕ, n = polygonalNumber (m + 1) x1 + polygonalNumber (m + 2) x2 + polygonalNumber (m + 3) x3

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

• A160324

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.