A230044 Nonnegative numbers k such that k plus a perfect square is a triangular number.

0, 1, 2, 3, 5, 6, 9, 10, 11, 12, 14, 15, 17, 19, 20, 21, 24, 27, 28, 29, 30, 32, 35, 36, 39, 41, 42, 44, 45, 46, 50, 51, 53, 54, 55, 56, 57, 62, 65, 66, 69, 71, 72, 74, 75, 77, 78, 80, 82, 84, 87, 89, 90, 91, 95, 96, 100, 101, 104, 105, 107, 109, 110, 111, 116, 117, 119, 120, 122, 126, 127, 128

Offset: 1

Author

Ralf Stephan, Oct 06 2013

Keywords

• nonn

Comments

Negative k are in A175035.

Numbers such that the Diophantine equation y^2 + y - 2x^2 = 2n, y > 0 has a solution. Empirically, solutions (x,y) don't exceed (5n,5n) for n < 10^5. Record quotients y/n are at n = 2, 3, 12, 45, 1225, 6806, ...

Conjecture: these are the sorted distinct terms of A064784.

n is in this sequence iff 8n+1 is in A035251, that is, every prime p == 3 or 5 (mod 8) dividing 8n+1 appears to an even power. - Max Alekseyev, Oct 14 2013

Examples

			28 is triangular, and 25 is a square <= 28, and 28-25=3, so 3 is in sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A035251, A064784, A175035.

Programs

PARI

B=bnfinit(z^2-8); is(n)=#bnfisintnorm(B,8*n+1) \\ Max Alekseyev, Oct 13 2013