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A115288 a(n) is the smallest number representable in exactly n ways as a sum of 2 triangular numbers and one square (each of them >= 0).

%I A115288 #6 Apr 28 2020 10:34:26
%S A115288 0,1,4,7,10,16,22,25,64,46,70,67,92,85,160,115,106,136,200,157,190,
%T A115288 172,256,235,568,277,370,337,400,367,340,550,556,442,1102,445,472,631,
%U A115288 610,535,682,697,652,1075,956,850,1984,865,1172,997,862,1081,1462,1135,1060
%N A115288 a(n) is the smallest number representable in exactly n ways as a sum of 2 triangular numbers and one square (each of them >= 0).
%e A115288 a(4)=7 since 7 can be expressed in 4 ways, 7= T(3)+T(1)+0^2 = T(3)+T(0)+1^2 = T(2)+T(0)+2^2 = T(2)+T(2)+1^2 and none of the numbers from 0 to 6 can be expressed in 4 ways.
%p A115288 a := [seq(0,n=0..100)] ;
%p A115288 for k from 0 do
%p A115288     a330861 := A330861(k) ;
%p A115288     if a330861 <= nops(a) then
%p A115288         if op(a330861,a) = 0 then
%p A115288             a := subsop(a330861=k,a) ;
%p A115288             print(a) ;
%p A115288         end if;
%p A115288     end if;
%p A115288     if not member(0,[op(2..nops(a),a)]) then
%p A115288         break;
%p A115288     end if;
%p A115288 end do: # _R. J. Mathar_, Apr 28 2020
%t A115288 V=Table[0, {i, 2500}]; T[n]:=n(n+1)/2; Do[a=T[i]+T[j]+k^2;If[a<2500, V[[a+1]]++ ], {i, 0, 71}, {j, 0, i}, {k, 0, 50}]; Table[Position[V, z][[1, 1]]-1, {z, 60}]
%Y A115288 Cf. A115289, A000437, A061262, A330861.
%K A115288 nonn
%O A115288 1,3
%A A115288 _Giovanni Resta_, Jan 19 2006