A242662 Nonnegative integers of the form x^2 + 4xy - 3y^2.

0, 1, 2, 4, 8, 9, 16, 18, 21, 25, 29, 32, 36, 37, 42, 49, 50, 53, 57, 58, 64, 72, 74, 81, 84, 93, 98, 100, 106, 109, 113, 114, 116, 121, 128, 133, 137, 141, 144, 148, 149, 162, 168, 169, 177, 186, 189, 193, 196, 197, 200, 212, 217, 218, 225, 226, 228, 232, 233, 242, 249, 256, 261, 266, 274, 277, 281, 282, 288, 289, 296, 298

Offset

0,3

Comments

Discriminant = 28.

Also nonnegative integers of the form x^2 - 7y^2. - Colin Barker, Sep 29 2014

Also nonnegative integers of the form x^2 + bxy + cy^2 where b = -2n, c = n^2 - 7, for integer n. This includes both forms above: x^2 + 4xy - 3y^2 with n = -2 and x^2 - 7y^2 with n = 0. - Klaus Purath, Jan 14 2023

For the subsequence of numbers that are properly represented see A358946. - Wolfdieter Lang, Jan 18 2023

Proof for the proper equivalence of the above given family of forms F(n) = [1, -2*n, n^2 -7], for integer n, with the reduced principal form of discriminant 28, namely F_p = [1, 4, -3] given in the name: In matrix form MF(n) = Matrix([[1, -n], [-n, n^2 -7]]) = R(n)^T*MF_p(n)*R(n), with MF_p(n) = Matrix([[1, 2], [2, -3]]) and R(n) = Matrix([[1, -(n+2)], [0, 1]]) (T for transposed). - Wolfdieter Lang, Jan 20 2023

Links

R. J. Mathar, Table of n, a(n) for n = 0..1184

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

Mathematica

Reap[For[n = 0, n <= 300, n++, If[Reduce[x^2 + 4*x*y - 3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]]

Crossrefs

Primes = A141172.

Cf. A242666, A358946.

Sequence in context: A155562 A048715 A336232 * A335851 A028982 A320137

Adjacent sequences: A242659 A242660 A242661 * A242663 A242664 A242665

Keyword

nonn

Author

N. J. A. Sloane, May 31 2014

Status

approved