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

1,1

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.

Links

Table of n, a(n) for n=1..50.

Example

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.

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. *)

Crossrefs

Cf. A067274.

Sequence in context: A070987 A168286 A079458 * A290378 A104537 A128018

Adjacent sequences: A281911 A281912 A281913 * A281915 A281916 A281917

Keyword

nonn

Author

Lorenz H. Menke, Jr., Feb 02 2017

Status

approved