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%I #13 Oct 02 2017 08:03:31
%S 1,1,1,1,1,1,1,1,2,2,0,1,2,1,2,3,0,1,3,1,2,3,1,1,1,1,3,3,1,2,1,2,3,1,
%T 2,4,2,1,3,2,2,3,2,1,2,2,2,3,3,2,1,2,2,3,5,2,2,2,2,3,4,2,2,4,1,3,2,1,
%U 4,3,2,2,5,2,4,3,0,4,2,1,3,6,3,3,3,1,5,2,3,5,2,2,3,3,1,5,3,1,3,3,4
%N Number of ways to write n as the sum of three unordered generalized heptagonal numbers.
%C Conjecture: (i) a(n) > 0 except for n = 10, 16, 76, 307.
%C (ii) For any integer m > 2 not divisible by 4, each sufficiently large integer n can be written as the sum of three generalized m-gonal numbers.
%C In 1994 R. K. Guy noted that none of 10, 16 and 76 can be written as the sum of three generalized heptagonal numbers.
%H Zhi-Wei Sun, <a href="/A256171/b256171.txt">Table of n, a(n) for n = 0..10000</a>
%H R. K. Guy, <a href="http://www.jstor.org/stable/2324367">Every number is expressible as the sum of how many polygonal numbers?</a>, Amer. Math. Monthly 101 (1994), 169-172.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1503.03743">A result similar to Lagrange's theorem</a>, arXiv:1503.03743 [math.NT], 2015.
%e a(157) = 1 since 157 = 3*(5*3-3)/2 + (-3)*(5*(-3)-3)/2 + 7*(5*7-3)/2.
%e a(748) = 1 since 748 = 0*(5*0-3)/2 + 0*(5*0-3)/2 + (-17)*(5*(-17)-3)/2.
%t T[n_]:=Union[Table[x(5x-3)/2, {x, -Floor[(Sqrt[40n+9]-3)/10], Floor[(Sqrt[40n+9]+3)/10]}]]
%t L[n_]:=Length[T[n]]
%t Do[r=0;Do[If[Part[T[n],x]>n/3,Goto[aa]];Do[If[Part[T[n],x]+2*Part[T[n],y]>n,Goto[bb]];
%t If[MemberQ[T[n], n-Part[T[n],x]-Part[T[n],y]]==True,r=r+1];
%t Continue,{y,x,L[n]}];Label[bb];Continue,{x,1,L[n]}];Label[aa];Print[n," ",r];Continue, {n,0,100}]
%Y Cf. A085787, A255934.
%K nonn
%O 0,9
%A _Zhi-Wei Sun_, Mar 17 2015
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