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

1,3

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

Links

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

Example

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

Prog

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

Crossrefs

Cf. A035251, A064784, A175035.

Sequence in context: A131292 A276476 A069880 * A329269 A069861 A379774

Adjacent sequences: A230041 A230042 A230043 * A230045 A230046 A230047

Keyword

nonn

Author

Ralf Stephan, Oct 06 2013

Status

approved