A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.

0, 2, 5, 8, 17, 18, 20, 32, 33, 42, 45, 50, 53, 68, 72, 77, 80, 98, 105, 113, 122, 125, 128, 132, 137, 153, 162, 168, 170, 173, 177, 180, 197, 200, 212, 213, 218, 233, 242, 245, 257, 258, 272, 288, 293, 297, 305, 308, 317, 320, 330, 338, 353, 357, 362, 378

Offset: 1

Author

N. J. A. Sloane, Jun 05 2014

Keywords

• nonn

Comments

Discriminant 60.

Nonnegative numbers of the form 5x^2 - 3y^2. - Jon E. Schoenfield, Jun 03 2022

From Klaus Purath, Jul 26 2023: (Start)

Nonnegative integers k such that 3x^2 - 5y^2 + k = 0 has integer solutions.

Also nonnegative integers of the form 2x^2 + (4m+2)xy + (2m^2+2m-7)y^2 for integers m. This includes the form in the name with m = 1.

Also nonnegative integers of the form 5x^2 + 10mxy + (5m^2-3)y^2 for integers m. This includes the form from Jon E. Schoenfield above with m = 0.

There are no squares in this sequence. Even powers of terms as well as products of an even number of terms belong to A243188.

Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the form 5x^2 - 3y^2 by the following identity: (na^2 - kb^2)(nc^2 - kd^2)(ne^2 - kf^2) = n[a(nce + kdf) + bk(cf + de)]^2 - k[na(cf + de) + b(nce + kdf)]^2 for all a, b, c, d, e, f, k, n in R. This can be verified by expanding both sides of the equation.

(End)

Links

• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Primes: A141303. Cf. A243188, A107152, A237606, A141302, A243190, A141304.

Programs

Mathematica

Reap[For[n = 0, n <= 200, n++, If[Reduce[2*x^2 + 6*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

Extensions

0 prepended and more terms from Colin Barker, Apr 07 2015