A067274 -id:A067274 - cp's OEIS frontend

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

Lorenz H. Menke, Jr., Feb 02 2017

nonn

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.

			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.

Cf. A067274.

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. *)

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.

Original entry on oeis.org

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

Lorenz H. Menke, Jr., Feb 02 2017

nonn

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.

			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.

Cf. A067274.

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. *)

A330407 Number of ordered integer pairs (b,c) with -n <= b <= n and -n <= c <= n such that both roots of x^2 + b*x + c = 0 are distinct integers.

Original entry on oeis.org

0, 3, 7, 13, 20, 26, 36, 42, 52, 59, 69, 75, 89, 95, 105, 115, 126, 132, 146, 152, 166, 176, 186, 192, 210, 217, 227, 237, 251, 257, 275, 281, 295, 305, 315, 325, 344, 350, 360, 370, 388, 394, 412, 418, 432, 446, 456, 462, 484, 491, 505, 515, 529, 535, 553, 563, 581

Offset: 0

Alexander Piperski, Jan 25 2020

nonn

			For n = 1, the a(1) = 3 equations are x^2 - x = 0, x^2 + x = 0, and x^2 - 1 = 0.
For n = 2, the a(2) = 7 equations are the 3 equations listed above and x^2 - 2x = 0, x^2 + 2x = 0, x^2 - x - 2 = 0, and x^2 + x - 2 = 0.

Cf. A001650, A067274.

ok[b_, c_] := Block[{d = b^2 - 4 c}, d > 0 && IntegerQ@ Sqrt@ d];  a[n_] := Sum[ Boole@ ok[b, c], {b, -n, n}, {c, -n, n}]; Array[a, 57, 0] (* Giovanni Resta, Jan 28 2020 *)

isok(b,c) = (b^2 > 4*c) && issquare(b^2-4*c);
a(n) = sum(b=-n, n, sum(c=-n, n, isok(b,c))); \\ Michel Marcus, Jan 28 2020

a(n) = A067274(n) - A001650(n+1) for n > 1.

a(0)=0 prepended by Michel Marcus, Jan 30 2020

cp's OEIS Frontend

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.

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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.

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A330407 Number of ordered integer pairs (b,c) with -n <= b <= n and -n <= c <= n such that both roots of x^2 + b*x + c = 0 are distinct integers.

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