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

Author

Zhi-Wei Sun, May 08 2009

Keywords

• nonn

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,...).

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..50000
• B. K. Oh and Z. W. 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).
• Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

Crossrefs

Cf. A000290, A000326, A160324, A160325.

Programs

Mathematica

SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n-y^2-(3z^2-z)/2],1,0], {y,0,Sqrt[n]},{z,0,Sqrt[n-y^2]}] Do[Print[n," ", RN[n]],{n,0,50000}]

Formula

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