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
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Programs
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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
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