

A160326


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


10



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

In April 2009, ZhiWei Sun conjectured that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. The GaussLegendre 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

ZhiWei 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), 964969.
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.
ZhiWei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.
ZhiWei 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], 20092015.


FORMULA

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


EXAMPLE

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


MATHEMATICA

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


CROSSREFS

Cf. A000290, A000326, A160324, A160325.
Sequence in context: A089680 A309507 A306690 * A213662 A213657 A215596
Adjacent sequences: A160323 A160324 A160325 * A160327 A160328 A160329


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 08 2009


STATUS

approved



