A281914 - OEIS

%I #13 Feb 09 2017 10:44:13

%S 2,8,8,16,24,24,34,46,46,60,72,74,86,100,104,122,132,132,142,164,168,

%T 182,192,200,214,228,228,250,260,268,278,300,304,318,336,340,350,364,

%U 368,398,408,416,426,448,452,466,476,488,502,524

%N 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.

%C 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.

%e The four quadratics for a(2)=8 and their roots are as follows:

%e 3*x^2 + 2*x + 0 = x(2 + 3*x);         x =  0, x = -2/3.

%e 3*x^2 + 2*x - 1 = (1 + x)(- 1 + 3*x); x = -1, x = +1/3.

%e 3*x^2 + 1*x + 0 = x(1 + 3*x);         x =  0, x = -1/3.

%e 3*x^2 + 1*x - 2 = (1 + x)(- 2 + 3*x); x = -1, x = +2/3.

%e 3*x^2 - 1*x + 0 = x(- 1 + 3*x);       x =  0, x = +1/3.

%e 3*x^2 - 1*x - 2 = (- 1 + x)(2 + 3*x); x = +1, x = -2/3.

%e 3*x^2 - 2*x + 0 = x(- 2 + 3*x);       x =  0, x = +2/3.

%e 3*x^2 - 2*x - 1 = (- 1 + x)(1 + 3*x); x = +1, x = -1/3.

%e There is one case where there is an overall factor of 3 which is counted in series A067274.

%t a[n_] :=

%t 2 (2 + Floor[(n + 1)/3] + Floor[(n - 1)/3] + Floor[(n + 2)/3] +

%t      Floor[(n - 2)/3]) +

%t   2 (KroneckerDelta[4, If[n == 4, 4, 0]] -

%t      KroneckerDelta[8, If[n == 8, 8, 0]] -

%t      KroneckerDelta[9, If[n == 9, 9, 0]] -

%t      KroneckerDelta[10, If[n == 10, 10, 0]] -

%t      KroneckerDelta[12, If[n == 12, 12, 0]]) +

%t   If[n >= 4,

%t    2 (-4 - 2 n - 2 Floor[n/2] + Floor[2 (n + 1)/3] +

%t       2 Sum[Length[Divisors[k]], {k, n}] -

%t       2 Sum[Length[Divisors[k]], {k, Floor[n/3]}]), 0];

%t (* The KroneckerDelta is a special case correction term. *)

%t a[1] = 2; (* Extends the a[n] series by direct count. *)

%Y Cf. A067274.

%K nonn

%O 1,1

%A _Lorenz H. Menke, Jr._, Feb 02 2017