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%I #16 Sep 10 2018 13:17:48
%S 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,
%T 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,
%U 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
%N Number of ways to express n=0,1,2,... as the sum of a triangular number, an even square and a pentagonal number.
%C 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).
%H Zhi-Wei Sun, <a href="/A160325/b160325.txt">Table of n, a(n) for n = 0..60000</a>
%H B. K. Oh and Z. W. Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2008.10.002">Mixed sums of squares and triangular numbers (III)</a>, J. Number Theory 129(2009), 964-969.
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.
%H Zhi-Wei Sun, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;48c9be36.0905">Various new conjectures involving polygonal numbers and primes</a> (a message to Number Theory List), 2009.
%H Zhi-Wei Sun, <a href="http://math.nju.edu.cn/~zwsun/MSPT.htm">Mixed Sums of Primes and Other Terms (a webpage)</a>.
%H Z. W. Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, preprint, arXiv:0905.0635 [math.NT], 2009-2015.
%F a(n) = |{<x,y,z>: x,y,z=0,1,2,... & x(x+1)/2+4y^2+(3z^2-z)/2}|.
%e For n=15 the a(15)=5 solutions are 3+0+12, 6+4+5, 10+0+5, 10+4+1, 15+0+0.
%t 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}]
%Y Cf. A000217, A000290, A000326, A160324.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, May 08 2009
%E More terms copied from author's b-file by _Hagen von Eitzen_, Jul 20 2009
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