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

%I A193773 #17 Dec 23 2018 20:04:06
%S A193773 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,
%T A193773 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,
%U A193773 1,1,2,2,2,3,1,1,3,2,1,2,2,1,4,1,2,3
%N A193773 Number of ways to write n as 2*x*y - x - y with 1 <= x <= y.
%C A193773 a(A005097(n)) = 1; for n > 1: a(A047845(n)) > 1. - _Reinhard Zumkeller_, Jan 02 2013
%C A193773 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
%H A193773 Reinhard Zumkeller, <a href="/A193773/b193773.txt">Table of n, a(n) for n = 0..10000</a>
%F A193773 a(n) = ceiling(A000005(2*n+1) / 2). - _Michael Somos_, Dec 23 2018
%e A193773 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
%t A193773 a[ n_] := If[ n < 0, 0, Ceiling[ DivisorSigma[0, 2 n + 1] / 2]]; (* _Michael Somos_, Dec 23 2018 *)
%o A193773 (Haskell)
%o A193773 a193773 n = length [() | x <- [1 .. n + 1],
%o A193773                          let (y,m) = divMod (x + n) (2 * x - 1),
%o A193773                          x <= y, m == 0]
%o A193773 (PARI) {a(n) = if(n < 0, 0, (numdiv(2*n+1) + 1)\2)}; /* _Michael Somos_, Dec 23 2018 */
%Y A193773 Cf. A000005, A005097, A047845, A125203.
%K A193773 nonn
%O A193773 0,5
%A A193773 _Reinhard Zumkeller_, Jan 02 2013