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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.
0
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.
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
KEYWORD
nonn
AUTHOR
STATUS
approved