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%I #37 Jul 21 2023 12:51:57
%S 1,1,1,3,1,2,3,1,3,1,3,3,2,4,2,3,3,3,4,3,2,5,1,2,4,3,5,4,5,4,4,3,6,3,
%T 3,2,5,2,3,7,3,7,2,6,3,5,6,7,2,4,6,3,7,2,8,4,2,6,6,3,8,3,4,6,3,7,5,6,
%U 7,4,6,9,5,6,4,4,3,4,9,5,6
%N Number of ways to write n as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z nonnegative integers.
%C Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 2, 4, 7, 9, 22.
%C It was proved in arXiv:1502.03056 that each n = 0,1,2,... can be written as x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2 with x,y,z integers. The author would like to offer 135 US dollars as the prize for the first proof of the conjecture that a(n) is always positive.
%C See over 400 similar conjectures in the linked a-file.
%H Zhi-Wei Sun, <a href="/A287616/b287616.txt">Table of n, a(n) for n = 0..10000</a>
%H Zhi-Wei Sun, <a href="/A287616/a287616_3.txt">List of conjectural tuples (a,b,c,d,e,f) with {x*(ax+b)/2 + y*(cy+d)/2 + z*(ez+f)/2: x,y,z = 0,1,2,...} = {0,1,2,...}</a>
%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://www.sciengine.com/SCM/doi/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Sci. China Math. 58 (2015), No. 7, 1367-1396.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums x(ax+b)/2+y(cy+d)/2+z(ez+f)/2</a>, arXiv:1502.03056 [math.NT], 2015-2017.
%e a(4) = 1 since 4 = 1*(1+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
%e a(7) = 1 since 7 = 0*(0+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.
%e a(9) = 1 since 9 = 3*(3+1)/2 + 0*(3*0+1)/2 + 1*(5*1+1)/2.
%e a(22) = 1 since 22 = 5*(5+1)/2 + 2*(3*2+1)/2 + 0*(5*0+1)/2.
%t TQ[n_]:=TQ[n]=IntegerQ[Sqrt[8n+1]];
%t Do[r=0;Do[If[TQ[n-x(3x+1)/2-y(5y+1)/2],r=r+1],{x,0,(Sqrt[24n+1]-1)/6},{y,0,(Sqrt[40(n-x(3x+1)/2)+1]-1)/10}];Print[n," ",r],{n,0,80}]
%Y Cf. A000217, A000290, A005449, A160324, A160325, A160326, A254668, A286944.
%K nonn
%O 0,4
%A _Zhi-Wei Sun_, May 27 2017
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