A243172 Nonnegative integers of the form x^2 + 3*x*y - 3*y^2 (discriminant 21).

0, 1, 4, 7, 9, 15, 16, 25, 28, 36, 37, 43, 49, 51, 60, 63, 64, 67, 79, 81, 85, 100, 105, 109, 112, 121, 123, 127, 135, 141, 144, 148, 151, 163, 169, 172, 175, 177, 193, 196, 204, 205, 211, 225, 235, 240, 249, 252, 256, 259, 267, 268, 277, 289, 295, 301, 303, 316, 324, 331, 333, 337, 340, 343, 357, 361, 373, 375, 379, 387, 393, 400, 415, 420, 421, 436, 441, 445, 448, 457, 459, 463, 469, 484, 487, 492, 499

Offset: 1

Author

N. J. A. Sloane, Jun 01 2014

Keywords

• nonn

Comments

Also numbers representable as x^2 + 5*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018

Also numbers of the form x^2 - x*y - 5*y^2 with 0 <= x <= y (or x^2 + x*y - 5*y^2 with x, y nonnegative). - Jianing Song, Jul 31 2018

Also nonnegative numbers of the form 7*x^2 - 3*y^2. - Jon E. Schoenfield, Jun 03 2022

Links

• Gheorghe Coserea, Table of n, a(n) for n = 1..100001
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A031363.

Primes: A141159.

Programs

Mathematica

A={}; For[ n=0, n <= 300, n++,
    If[ Length[ Reduce[x^2 + 3 x y - 3 y^2 - n == 0, {x,y}, Integers]]>0, AppendTo[A,n]]]; A

PARI

\\ From Bill Allombert, Jun 04 2014. Since 21 is a fundamental discriminant, and the polynomial is unitary, the following code works:
B=bnfinit(x^2+3*x-3); select(n->#bnfisintnorm(B,n),[1..500])