A234336 Triangular numbers t such that both distances from t to two nearest squares are perfect squares.

0, 1, 45, 153, 325, 10440, 1385280, 2530125, 145462096, 253472356000, 896473314291600, 18598323060963360, 4923539323344237960, 27021247523935843321, 1779312917089890560241, 2355054824151326520405, 21328127890911040269960, 124797500891024855239125

Offset: 1

Author

Alex Ratushnyak, Dec 23 2013

Keywords

• nonn

Comments

Triangular numbers in A234334.

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.

The sequence of k's such that triangular(k) is in A234334 begins: 0, 1, 9, 17, 25, 144, 1664, 2249, 17056, 712000, ...

Examples

			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.
		

Crossrefs

Cf. A000217, A000290, A229909, A234334.

Programs

C

#include 
#include 
typedef unsigned long long U64;
U64 isSquare(U64 a) {
  U64 r = sqrt(a);
  return r*r==a;
}
int main() {
  for (U64 i=0; i<(1ULL<<32); ++i) {
    U64 n = i*(i+1)/2, r = sqrt(n);
    if (r*r==n && n)  --r;
    if (isSquare(n-r*r) && isSquare((r+1)*(r+1)-n))
      printf("%llu, ", n);
  }
  return 0;
}