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%I #15 Jan 01 2014 02:29:13
%S 0,1,45,153,325,10440,1385280,2530125,145462096,253472356000,
%T 896473314291600,18598323060963360,4923539323344237960,
%U 27021247523935843321,1779312917089890560241,2355054824151326520405,21328127890911040269960,124797500891024855239125
%N Triangular numbers t such that both distances from t to two nearest squares are perfect squares.
%C Triangular numbers in A234334.
%C Except a(1)=0, a(n) are triangular numbers t such that both t-x and y-t are perfect squares, where x and y are two nearest to k squares: x < t <= y.
%C The sequence of k's such that triangular(k) is in A234334 begins: 0, 1, 9, 17, 25, 144, 1664, 2249, 17056, 712000, ...
%e Triangular(9) = 45 is in the sequence because both 45-36=9 and 49-45=4 are perfect squares, where 36 and 49 are the two squares nearest to 45.
%o (C)
%o #include <stdio.h>
%o #include <math.h>
%o typedef unsigned long long U64;
%o U64 isSquare(U64 a) {
%o U64 r = sqrt(a);
%o return r*r==a;
%o }
%o int main() {
%o for (U64 i=0; i<(1ULL<<32); ++i) {
%o U64 n = i*(i+1)/2, r = sqrt(n);
%o if (r*r==n && n) --r;
%o if (isSquare(n-r*r) && isSquare((r+1)*(r+1)-n))
%o printf("%llu, ", n);
%o }
%o return 0;
%o }
%Y Cf. A000217, A000290, A229909, A234334.
%K nonn
%O 1,3
%A _Alex Ratushnyak_, Dec 23 2013
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