A330407 Number of ordered integer pairs (b,c) with -n <= b <= n and -n <= c <= n such that both roots of x^2 + b*x + c = 0 are distinct integers.

0, 3, 7, 13, 20, 26, 36, 42, 52, 59, 69, 75, 89, 95, 105, 115, 126, 132, 146, 152, 166, 176, 186, 192, 210, 217, 227, 237, 251, 257, 275, 281, 295, 305, 315, 325, 344, 350, 360, 370, 388, 394, 412, 418, 432, 446, 456, 462, 484, 491, 505, 515, 529, 535, 553, 563, 581

Offset: 0

Author

Alexander Piperski, Jan 25 2020

Keywords

• nonn

Examples

			For n = 1, the a(1) = 3 equations are x^2 - x = 0, x^2 + x = 0, and x^2 - 1 = 0.
For n = 2, the a(2) = 7 equations are the 3 equations listed above and x^2 - 2x = 0, x^2 + 2x = 0, x^2 - x - 2 = 0, and x^2 + x - 2 = 0.
		

Crossrefs

Cf. A001650, A067274.

Programs

Mathematica

ok[b_, c_] := Block[{d = b^2 - 4 c}, d > 0 && IntegerQ@ Sqrt@ d];  a[n_] := Sum[ Boole@ ok[b, c], {b, -n, n}, {c, -n, n}]; Array[a, 57, 0] (* Giovanni Resta, Jan 28 2020 *)

PARI

isok(b,c) = (b^2 > 4*c) && issquare(b^2-4*c);
a(n) = sum(b=-n, n, sum(c=-n, n, isok(b,c))); \\ Michel Marcus, Jan 28 2020

Formula

a(n) = A067274(n) - A001650(n+1) for n > 1.

Extensions

a(0)=0 prepended by Michel Marcus, Jan 30 2020