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%I #50 Apr 11 2015 09:20:39
%S 1,2,2,2,1,1,1,3,4,2,2,1,4,3,3,4,2,3,1,3,2,2,5,3,3,3,3,6,3,6,4,2,3,1,
%T 7,2,4,5,5,4,1,5,5,2,3,4,4,5,5,5,3,5,7,6,4,3,1,6,6,8,5,3,6,4,7,4,2,6,
%U 5,5,3,4,8,3,3,3,6,6,7,9,6,2,5,6,7,7,4,6,6,7,5,3,10,6,3,4,5,7,3,10,7
%N Number of ordered ways to write n as the sum of a pentagonal number, a second pentagonal number and a generalized decagonal number.
%C Conjecture: a(n) > 0 for all n. Also, for any ordered pair (k,m) among (5,7), (5,9), (5,13), (6,5), (6,7), (7,5), each nonnegative integer n can be written as the sum of a k-gonal number, a second k-gonal number and a generalized m-gonal number.
%C See also the author's similar conjectures in A254574, A254631, A255916 and the two linked papers.
%H Zhi-Wei Sun, <a href="/A253187/b253187.txt">Table of n, a(n) for n = 0..10000</a>
%H 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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums a*x^2+b*y^2+f(z), a*T_x+b*T_y+f(z) and a*T_x+b*y^2+f(z)</a>, arXiv:1502.03056 [math.NT], 2015.
%e a(33) = 1 since 33 = 0*(3*0-1)/2 + 4*(3*4+1)/2 + 1*(4*1+3).
%e a(56) = 1 since 56 = 4*(3*4-1)/2 + 2*(3*2+1)/2 + 3*(4*3+3).
%t DQ[n_]:=IntegerQ[Sqrt[16n+9]]
%t Do[r=0;Do[If[DQ[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 Print[n," ",r];Continue,{n,0,100}]
%Y Cf. A000326, A000384, A000566, A005449, A014105, A074377, A085787, A147875, A254574, A254631.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Apr 07 2015
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