A160325 Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.

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

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. 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

Zhi-Wei 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), 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,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/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(n-4y^2-(3z^2-z)/2)+1], 1, 0], {y, 0, Sqrt[n/4]}, {z, 0, Sqrt[n-4y^2]}] Do[Print[n, " ", RN[n]], {n, 0, 60000}]

Crossrefs

Cf. A000217, A000290, A000326, A160324.

Sequence in context: A343026 A113136 A156267 * A054989 A051631 A073725

Adjacent sequences: A160322 A160323 A160324 * A160326 A160327 A160328

Keyword

nonn

Author

Zhi-Wei Sun, May 08 2009

Extensions

More terms copied from author's b-file by Hagen von Eitzen, Jul 20 2009

Status

approved