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
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.
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
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 08 2009
STATUS
approved