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

0,2

Links

Table of n, a(n) for n=0..56.

Formula

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

Example

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.

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 *)

Prog

(Python 3.7) [sum([1 for b in range(-n, n+1) for c in range(-n, n+1) if b**2-4*c > 0 and int((-b+(b**2-4*c)**0.5)/2) == (-b+(b**2-4*c)**0.5)/2]) for n in range(0, 101)]
(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

Crossrefs

Cf. A001650, A067274.

Sequence in context: A106080 A187819 A310266 * A014283 A341299 A294398

Adjacent sequences: A330404 A330405 A330406 * A330408 A330409 A330410

Keyword

nonn

Author

Alexander Piperski, Jan 25 2020

Extensions

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

Status

approved