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.

4, 4, 12, 12, 22, 24, 36, 36, 50, 54, 64, 68, 78, 82, 100, 100, 110, 118, 128, 132, 150, 154, 164, 168, 182, 186, 204, 208, 218, 230, 240, 240, 258, 262, 280, 288, 298, 302, 320, 324, 334, 346, 356, 360, 386, 390, 400, 404, 418, 426

Offset

1,1

Comments

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.

Links

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

Example

The four quadratics for a(2)=4 and their roots are as follows:
2*x^2 + 1*x + 0 = x(1 + 2*x);         x =  0, x = -1/2.
2*x^2 + 1*x - 1 = (1 + x)(- 1 + 2*x); x = -1, x = +1/2.
2*x^2 - 1*x + 0 = x(- 1 + 2*x);       x =  0, x = +1/2.
2*x^2 - 1*x - 1 = (- 1+ x)(1 + 2*x);  x = +1, x = -1/2.
There are nine cases where there is an overall factor of 2 which are counted in series A067274.

Mathematica

a[n_] := If[n >= 3,
   2 (-2 - 2 n + Floor[(n + 1)/2] +
      2 Sum[Length[Divisors[k]], {k, n}] -
      2 Sum[Length[Divisors[k]], {k, Floor[n/2]}]), 0] +
  4 Floor[(n + 1)/2] - 2 KroneckerDelta[6, If[n == 6, 6, 0]];
(* The KroneckerDelta is a special case correction term. *)
a[1] = 4; (* Extends the a[n] series by direct count. *)

Crossrefs

Cf. A067274.

Sequence in context: A206488 A273491 A168398 * A178776 A282503 A323189

Adjacent sequences: A281910 A281911 A281912 * A281914 A281915 A281916

Keyword

nonn

Author

Lorenz H. Menke, Jr., Feb 02 2017

Status

approved