A287616 Number of ways to write n as x*(x+1)/2 + y*(3*y+1)/2 + z*(5*z+1)/2 with x,y,z nonnegative integers.

1, 1, 1, 3, 1, 2, 3, 1, 3, 1, 3, 3, 2, 4, 2, 3, 3, 3, 4, 3, 2, 5, 1, 2, 4, 3, 5, 4, 5, 4, 4, 3, 6, 3, 3, 2, 5, 2, 3, 7, 3, 7, 2, 6, 3, 5, 6, 7, 2, 4, 6, 3, 7, 2, 8, 4, 2, 6, 6, 3, 8, 3, 4, 6, 3, 7, 5, 6, 7, 4, 6, 9, 5, 6, 4, 4, 3, 4, 9, 5, 6

Offset

0,4

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2, 4, 7, 9, 22.

It was proved in arXiv:1502.03056 that each n = 0,1,2,... can be written as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z integers. The author would like to offer 135 US dollars as the prize for the first proof of the conjecture that a(n) is always positive.

See over 400 similar conjectures in the linked a-file.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Yichuan Cao, Dakai Guo, Ruichen Qiu, Ruyong Feng, and Xiao-Shan Gao, Every Nonnegative Integer Is a Sum of a Triangular, a Pentagonal, and a Heptagonal Number, arXiv:2606.26035 [math.NT], 2026. See pp. 1-2, 12 (Lean formalization).

Zhi-Wei Sun, List of conjectural tuples (a,b,c,d,e,f) with {x*(ax+b)/2 + y*(cy+d)/2 + z*(ez+f)/2: x,y,z = 0,1,2,...} = {0,1,2,...}

Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127 (2007), 103-113.

Zhi-Wei Sun, On universal sums of polygonal numbers, Sci. China Math. 58 (2015), No. 7, 1367-1396.

Zhi-Wei Sun, Universal sums of three quadratic polynomials, arXiv:1502.03056 [math.NT], 2015-2020.

Example

a(4) = 1 since 4 = 1*(1+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
a(7) = 1 since 7 = 0*(0+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.
a(9) = 1 since 9 = 3*(3+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
a(22) = 1 since 22 = 5*(5+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.

Mathematica

TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
Do[r=0; Do[If[TQ[n-x(3x+1)/2-y(5y+1)/2], r=r+1], {x, 0, (Sqrt[24n+1]-1)/6}, {y, 0, (Sqrt[40(n-x(3x+1)/2)+1]-1)/10}]; Print[n, " ", r], {n, 0, 80}]

Crossrefs

Cf. A000217, A000290, A005449, A160324, A160325, A160326, A254668, A286944.

Sequence in context: A079723 A080511 A132399 * A371101 A081485 A307693

Adjacent sequences: A287613 A287614 A287615 * A287617 A287618 A287619

Keyword

nonn,changed

Author

Zhi-Wei Sun, May 27 2017

Status

approved