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

1,3

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, ...

Links

Table of n, a(n) for n=1..18.

Example

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.

Prog

(C)
#include <stdio.h>
#include <math.h>
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;
}

Crossrefs

Cf. A000217, A000290, A229909, A234334.

Sequence in context: A044758 A053743 A347874 * A173371 A288669 A305069

Adjacent sequences: A234333 A234334 A234335 * A234337 A234338 A234339

Keyword

nonn

Author

Alex Ratushnyak, Dec 23 2013

Status

approved