

A160325


Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.


16



1, 2, 1, 1, 2, 3, 3, 2, 2, 1, 3, 3, 2, 1, 1, 5, 3, 3, 2, 4, 3, 2, 6, 2, 2, 2, 5, 4, 3, 3, 1, 4, 4, 3, 1, 1, 5, 7, 5, 3, 4, 6, 4, 3, 4, 5, 2, 3, 3, 5, 4, 5, 5, 2, 6, 2, 5, 5, 5, 3, 3, 6, 3, 2, 5, 4, 6, 6, 3, 3, 6, 9, 6, 5, 4, 5, 5, 6, 2, 7, 4, 3, 6, 6, 4, 2, 7, 7, 3, 3, 4, 5, 8, 5, 5, 5, 8, 4, 2, 4, 6, 6, 7, 6, 4
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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. It is known that any positive integer can be written as the sum of a triangular number, a square and an even square (or an odd square).


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..60000
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,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2z)/2}.


EXAMPLE

For n=15 the a(15)=5 solutions are 3+0+12, 6+4+5, 10+0+5, 10+4+1, 15+0+0.


MATHEMATICA

SQ[x_]:=x>1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[8(n4y^2(3z^2z)/2)+1], 1, 0], {y, 0, Sqrt[n/4]}, {z, 0, Sqrt[n4y^2]}] Do[Print[n, " ", RN[n]], {n, 0, 60000}]


CROSSREFS

Cf. A000217, A000290, A000326, A160324.
Sequence in context: A326281 A113136 A156267 * A054989 A051631 A073725
Adjacent sequences: A160322 A160323 A160324 * A160326 A160327 A160328


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 08 2009


EXTENSIONS

More terms copied from author's bfile by Hagen von Eitzen, Jul 20 2009


STATUS

approved



