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A256855 Number of ordered ways to write n as x*(3*x-1)/2 + y*(3*y+1)/2 + z*(3*z+1), where x and y are nonnegative integers and z is an integer.

%I A256855 #6 Apr 11 2015 09:21:11
%S A256855 1,1,2,2,2,3,1,4,1,3,2,3,4,1,4,3,5,4,3,3,2,3,5,1,5,2,7,3,3,6,4,7,3,4,
%T A256855 2,3,4,7,4,5,4,6,5,3,5,6,4,4,3,4,5,8,8,4,5,6,5,5,4,7,5,6,7,3,7,4,7,5,
%U A256855 6,2,5,8,10,3,4,6,6,7,4,9,5,10,10,5,4,6,6,8,4,8,3,6,11,5,8,4,10,4,2,8,5
%N A256855 Number of ordered ways to write n as x*(3*x-1)/2 + y*(3*y+1)/2 + z*(3*z+1), where x and y are nonnegative integers and z is an integer.
%C A256855 Conjecture: a(n) > 0 for all n. Also, any nonnegative integer can be written as x*(3*x-1)/2 + y*(3*y-1) + z*(3*z+1)/2 with x,y nonnegative integers and z an integer, and each n = 0,1,2,... can be expressed as x*(3*x+1)/2 + 3*y*(3*y+1)/2 + z*(3*z+1)/2 with x,y nonnegative integers and z an integer.
%H A256855 Zhi-Wei Sun, <a href="/A256855/b256855.txt">Table of n, a(n) for n = 0..10000</a>
%H A256855 Zhi-Wei Sun, <a href="http://arxiv.org/abs/0905.0635">On universal sums of polygonal numbers</a>, arXiv:0905.0635 [math.NT], 2009-2015.
%e A256855  a(8) = 1 since 8 = 1*(3*1-1)/2 + 2*(3*2+1)/2 + 0*(3*0+1).
%e A256855 a(23) = 1 since 23 = 3*(3*3-1)/2 + 2*(3*2+1)/2 + 1*(3*1+1).
%t A256855 PQ[n_]:=IntegerQ[Sqrt[12n+1]]
%t A256855 Do[r=0;Do[If[PQ[n-x(3x-1)/2-y(3y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]+1)/6},{y,0,(Sqrt[24(n-x(3x-1)/2)+1]-1)/6}];
%t A256855 Print[n," ",r];Continue,{n,0,100}]
%Y A256855 Cf. A000326, A001318, A005449, A253187, A254573, A254574.
%K A256855 nonn
%O A256855 0,3
%A A256855 _Zhi-Wei Sun_, Apr 11 2015