A260195 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.

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

Offset: 0

Author

Michael Somos, Jul 18 2015

Keywords

• nonn

Comments

Same as A238872 except a(0) = 0.

Examples

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

Links

• Wikipedia, Hurwitz class number.

Crossrefs

Cf. A130695, A238872, A259825.

Programs

Mathematica

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];
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];

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};

Formula

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.