A231354 Least k such that n - triangular(k) is a square, or -1 if no such k exists.

0, 0, 1, 2, 0, 1, 3, 2, -1, 0, 1, 4, 2, -1, 4, 3, 0, 1, -1, 2, -1, 6, 3, -1, 5, 0, 1, -1, 2, 7, 6, 3, 7, -1, -1, 4, 0, 1, -1, 2, 5, -1, 3, -1, 7, 8, 4, -1, -1, 0, 1, 5, 2, 7, 9, 3, 10, 6, -1, 4, -1, 8, -1, -1, 0, 1, 11, 2, -1, -1, 3, 10, 8, -1, 4, 11, -1, 7, 12, 5, 10, 0

Offset: 0

Author

Alex Ratushnyak, Nov 08 2013

Keywords

• sign

Comments

Indices of -1's: A014134.

Indices of 0's: A000290.

Indices of 1's: A002522.

Crossrefs

Cf. A000217, A000290, A002522, A014134, A231353.

Programs

Python

import math
for n in range(333):
  for k in range(1000000):
    t = n - k*(k+1)/2
    if t<0:
      print('-1', end=', ')
      break
    r = int(math.sqrt(t))
    if r*r==t:
      print(str(k), end=', ')
      break