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
Keywords
Examples
For n=15 the a(15)=5 solutions are 3+0+12, 6+4+5, 10+0+5, 10+4+1, 15+0+0.
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.
Programs
-
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}]
Formula
a(n) = |{: x,y,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/2}|.
Extensions
More terms copied from author's b-file by Hagen von Eitzen, Jul 20 2009
Comments