A193773 Number of ways to write n as 2*x*y - x - y with 1 <= x <= y.

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 2, 2, 1, 3, 1, 1, 4, 1, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 2, 1, 2, 4, 1, 1, 2, 2, 2, 3, 1, 1, 3, 2, 1, 2, 2, 1, 4, 1, 2, 3

Offset: 0

Author

Reinhard Zumkeller, Jan 02 2013

Keywords

• nonn

Comments

a(A005097(n)) = 1; for n > 1: a(A047845(n)) > 1. - Reinhard Zumkeller, Jan 02 2013

Number of ways to write 2*n+1 as a difference of two squares. Note that 2*(2*x*y - x - y) + 1 = (2*x - 1) * (2*y - 1) = (y + x - 1)^2 - (y - x)^2. - Michael Somos, Dec 23 2018

Examples

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + x^8 + x^9 + 2*x^10 + ... - _Michael Somos_, Dec 23 2018
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000005, A005097, A047845, A125203.

Programs

Haskell

a193773 n = length [() | x <- [1 .. n + 1],
                         let (y,m) = divMod (x + n) (2 * x - 1),
                         x <= y, m == 0]

Mathematica

a[ n_] := If[ n < 0, 0, Ceiling[ DivisorSigma[0, 2 n + 1] / 2]]; (* Michael Somos, Dec 23 2018 *)

PARI

{a(n) = if(n < 0, 0, (numdiv(2*n+1) + 1)\2)}; /* Michael Somos, Dec 23 2018 */

Formula

a(n) = ceiling(A000005(2*n+1) / 2). - Michael Somos, Dec 23 2018