A260195 - OEIS

%I #3 Jul 18 2015 23:33:16

%S 0,1,1,3,2,3,3,4,3,6,4,3,5,6,4,9,5,3,7,7,5,9,6,6,8,9,5,9,8,6,10,6,5,

%T 15,8,9,10,7,7,12,10,3,11,15,7,15,8,6,13,12,9,12,9,9,14,12,7,15,12,6,

%U 15,13,6,21,12,12,13,6,11,15,15,9,14,12,8,24,10,9

%N Number of integer triples [x, y, z] such that 1 <= min(x,z), max(x,z) <= y, y^2 - (x^2 - x + z^2 - z) / 2 = n.

%C Same as A238872 except a(0) = 0.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hurwitz_class_number">Hurwitz class number</a>.

%F a(n) = A238872(n) unless n=0. a(2*n) = A130695(2*n) / 3. a(2*n + 1) = A130695(2*n + 1) = A259825(8*n + 3) / 4 = 3 * H(8*n + 3) where H() is the Hurwitz class number.

%e G.f. = x + x^2 + 3*x^3 + 2*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 3*x^8 + ...

%t a[ n_] := If[ OddQ[n], 1, 1/3] Length @ FindInstance[ {x >= 0, y >= 0, z >= 0, x y + y z + z x + x + y + z + 1 == n}, {x, y, z}, Integers, 10^9];

%t a[ n_] := Length @ FindInstance[ {1 <= y <= n, 1 <= x <= y, 1 <= z <= y, y^2 - (x^2 - x + z^2 - z) / 2 == n}, {x, y, z}, Integers, 10^9];

%o (PARI) {a(n) = my(c, t, i); for(k=1 + sqrtint(max(0, n-1)), n, forstep(j=1, min(2*k, sqrtint(t = 8*k^2 - 8*n + 2)), 2, if( issquare( t - j^2, &i) && i<=2*k, c++))); c};

%Y Cf. A130695, A238872, A259825.

%K nonn

%O 0,4

%A _Michael Somos_, Jul 18 2015