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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.

%I A281914 #13 Feb 09 2017 10:44:13
%S A281914 2,8,8,16,24,24,34,46,46,60,72,74,86,100,104,122,132,132,142,164,168,
%T A281914 182,192,200,214,228,228,250,260,268,278,300,304,318,336,340,350,364,
%U A281914 368,398,408,416,426,448,452,466,476,488,502,524
%N 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.
%C A281914 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 A281914 The four quadratics for a(2)=8 and their roots are as follows:
%e A281914 3*x^2 + 2*x + 0 = x(2 + 3*x);         x =  0, x = -2/3.
%e A281914 3*x^2 + 2*x - 1 = (1 + x)(- 1 + 3*x); x = -1, x = +1/3.
%e A281914 3*x^2 + 1*x + 0 = x(1 + 3*x);         x =  0, x = -1/3.
%e A281914 3*x^2 + 1*x - 2 = (1 + x)(- 2 + 3*x); x = -1, x = +2/3.
%e A281914 3*x^2 - 1*x + 0 = x(- 1 + 3*x);       x =  0, x = +1/3.
%e A281914 3*x^2 - 1*x - 2 = (- 1 + x)(2 + 3*x); x = +1, x = -2/3.
%e A281914 3*x^2 - 2*x + 0 = x(- 2 + 3*x);       x =  0, x = +2/3.
%e A281914 3*x^2 - 2*x - 1 = (- 1 + x)(1 + 3*x); x = +1, x = -1/3.
%e A281914 There is one case where there is an overall factor of 3 which is counted in series A067274.
%t A281914 a[n_] :=
%t A281914 2 (2 + Floor[(n + 1)/3] + Floor[(n - 1)/3] + Floor[(n + 2)/3] +
%t A281914      Floor[(n - 2)/3]) +
%t A281914   2 (KroneckerDelta[4, If[n == 4, 4, 0]] -
%t A281914      KroneckerDelta[8, If[n == 8, 8, 0]] -
%t A281914      KroneckerDelta[9, If[n == 9, 9, 0]] -
%t A281914      KroneckerDelta[10, If[n == 10, 10, 0]] -
%t A281914      KroneckerDelta[12, If[n == 12, 12, 0]]) +
%t A281914   If[n >= 4,
%t A281914    2 (-4 - 2 n - 2 Floor[n/2] + Floor[2 (n + 1)/3] +
%t A281914       2 Sum[Length[Divisors[k]], {k, n}] -
%t A281914       2 Sum[Length[Divisors[k]], {k, Floor[n/3]}]), 0];
%t A281914 (* The KroneckerDelta is a special case correction term. *)
%t A281914 a[1] = 2; (* Extends the a[n] series by direct count. *)
%Y A281914 Cf. A067274.
%K A281914 nonn
%O A281914 1,1
%A A281914 _Lorenz H. Menke, Jr._, Feb 02 2017