%I #10 Dec 26 2015 18:43:34
%S 81,106,181,211,256,276,331,361,381,406,456,471,531,556,601,606,616,
%T 631,666,681,706,718,731,781,786,856,861,871,906,931,939,956,981,991,
%U 1051,1056,1126,1131,1156,1186,1206,1231,1281,1296,1341,1381,1446,1456
%N Numbers that are the sum of two distinct nonzero triangular numbers in more than two ways.
%t r = 1456; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 2]
%t Module[{nn=60,trnos},trnos=Accumulate[Range[nn]];Select[Sort[Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,3,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* _Harvey P. Dale_, Dec 26 2015 *)
%Y Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).
%K nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Dec 02 2015