A160326 Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number.

1, 3, 3, 1, 2, 5, 4, 1, 1, 5, 6, 2, 1, 5, 5, 2, 4, 6, 5, 1, 3, 6, 5, 3, 1, 8, 8, 4, 2, 4, 8, 4, 5, 1, 4, 5, 4, 10, 6, 6, 5, 8, 6, 1, 3, 6, 6, 4, 6, 4, 7, 8, 8, 8, 5, 7, 4, 4, 6, 5, 6, 8, 7, 4, 8, 8, 6, 5, 4, 7, 7, 8, 7, 7, 8, 8, 8, 7, 3, 4, 12, 4, 4, 7, 3, 13, 12, 12, 5, 2, 12, 4, 5, 6, 6, 8, 10, 8, 3, 5, 11

Offset

0,2

Comments

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. The Gauss-Legendre theorem asserts that n is the sum of three squares if and only if it is not of the form 4^a(8b+7) (a,b=0,1,2,...).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..50000

Byeong-Kweon Oh and Zhi-Wei Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969.

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

Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.

Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).

Zhi-Wei Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Formula

a(n) = |{<x,y,z>: x,y=0,1,2,... & x^2+y^2+(3z^2-z)/2=n}|.

Example

For n=5 the a(5)=5 solutions are 0+0+5, 1+4+0, 4+1+0, 0+4+1, 4+0+1.

Mathematica

a = Compile[{{n, _Integer}}, Block[{c = 0}, Do[c += Boole[ Mod[ Sqrt[n - i (3 i - 1)/2 - j^2], 1] == 0], {i, 0, (1 + Sqrt[1 + 24 n])/6}, {j, 0, Sqrt[n - i (3 i - 1)/2]}]; c]]; Array[a, 101, 0] (* Robert G. Wilson v, Sep 04 2025 *)

Crossrefs

Cf. A000290, A000326, A160324, A160325.

Sequence in context: A394586 A309507 A306690 * A213662 A213657 A215596

Adjacent sequences: A160323 A160324 A160325 * A160327 A160328 A160329

Keyword

nonn

Author

Zhi-Wei Sun, May 08 2009

Status

approved