A281914 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 3x^2+bx+c = 0 are rational and b and are not both multiples of 3.

2, 8, 8, 16, 24, 24, 34, 46, 46, 60, 72, 74, 86, 100, 104, 122, 132, 132, 142, 164, 168, 182, 192, 200, 214, 228, 228, 250, 260, 268, 278, 300, 304, 318, 336, 340, 350, 364, 368, 398, 408, 416, 426, 448, 452, 466, 476, 488, 502, 524

Offset: 1

Author

Lorenz H. Menke, Jr., Feb 02 2017

Keywords

• nonn

Comments

We are not counting the cases where there is a possible overall factor of 3. When there is an overall factor of 3 we get the sequence A067274. These results have been proved and will appear in an upcoming paper.

Examples

			The four quadratics for a(2)=8 and their roots are as follows:
3*x^2 + 2*x + 0 = x(2 + 3*x);         x =  0, x = -2/3.
3*x^2 + 2*x - 1 = (1 + x)(- 1 + 3*x); x = -1, x = +1/3.
3*x^2 + 1*x + 0 = x(1 + 3*x);         x =  0, x = -1/3.
3*x^2 + 1*x - 2 = (1 + x)(- 2 + 3*x); x = -1, x = +2/3.
3*x^2 - 1*x + 0 = x(- 1 + 3*x);       x =  0, x = +1/3.
3*x^2 - 1*x - 2 = (- 1 + x)(2 + 3*x); x = +1, x = -2/3.
3*x^2 - 2*x + 0 = x(- 2 + 3*x);       x =  0, x = +2/3.
3*x^2 - 2*x - 1 = (- 1 + x)(1 + 3*x); x = +1, x = -1/3.
There is one case where there is an overall factor of 3 which is counted in series A067274.
		

Crossrefs

Cf. A067274.

Programs

Mathematica

a[n_] :=
2 (2 + Floor[(n + 1)/3] + Floor[(n - 1)/3] + Floor[(n + 2)/3] +
     Floor[(n - 2)/3]) +
  2 (KroneckerDelta[4, If[n == 4, 4, 0]] -
     KroneckerDelta[8, If[n == 8, 8, 0]] -
     KroneckerDelta[9, If[n == 9, 9, 0]] -
     KroneckerDelta[10, If[n == 10, 10, 0]] -
     KroneckerDelta[12, If[n == 12, 12, 0]]) +
  If[n >= 4,
   2 (-4 - 2 n - 2 Floor[n/2] + Floor[2 (n + 1)/3] +
      2 Sum[Length[Divisors[k]], {k, n}] -
      2 Sum[Length[Divisors[k]], {k, Floor[n/3]}]), 0];
(* The KroneckerDelta is a special case correction term. *)
a[1] = 2; (* Extends the a[n] series by direct count. *)