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A281913 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 2x^2+bx+c=0 are rational and b and c are not both even.

%I A281913 #11 Feb 09 2017 10:42:35
%S A281913 4,4,12,12,22,24,36,36,50,54,64,68,78,82,100,100,110,118,128,132,150,
%T A281913 154,164,168,182,186,204,208,218,230,240,240,258,262,280,288,298,302,
%U A281913 320,324,334,346,356,360,386,390,400,404,418,426
%N A281913 Number of ordered integer pairs (b,c), with -n<=b<=n, -n<=c<=n, such that both roots of 2x^2+bx+c=0 are rational and b and c are not both even.
%C A281913 We are not counting the cases where there is a possible overall factor of 2. When there is an overall factor of 2 we obtain the sequence A067274. These results have been proved and will appear in an upcoming paper.
%e A281913 The four quadratics for a(2)=4 and their roots are as follows:
%e A281913 2*x^2 + 1*x + 0 = x(1 + 2*x);         x =  0, x = -1/2.
%e A281913 2*x^2 + 1*x - 1 = (1 + x)(- 1 + 2*x); x = -1, x = +1/2.
%e A281913 2*x^2 - 1*x + 0 = x(- 1 + 2*x);       x =  0, x = +1/2.
%e A281913 2*x^2 - 1*x - 1 = (- 1+ x)(1 + 2*x);  x = +1, x = -1/2.
%e A281913 There are nine cases where there is an overall factor of 2 which are counted in series A067274.
%t A281913 a[n_] := If[n >= 3,
%t A281913    2 (-2 - 2 n + Floor[(n + 1)/2] +
%t A281913       2 Sum[Length[Divisors[k]], {k, n}] -
%t A281913       2 Sum[Length[Divisors[k]], {k, Floor[n/2]}]), 0] +
%t A281913   4 Floor[(n + 1)/2] - 2 KroneckerDelta[6, If[n == 6, 6, 0]];
%t A281913 (* The KroneckerDelta is a special case correction term. *)
%t A281913 a[1] = 4; (* Extends the a[n] series by direct count. *)
%Y A281913 Cf. A067274.
%K A281913 nonn
%O A281913 1,1
%A A281913 _Lorenz H. Menke, Jr._, Feb 02 2017