This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A243189 #22 Aug 26 2023 15:06:42
%S A243189 0,2,5,8,17,18,20,32,33,42,45,50,53,68,72,77,80,98,105,113,122,125,
%T A243189 128,132,137,153,162,168,170,173,177,180,197,200,212,213,218,233,242,
%U A243189 245,257,258,272,288,293,297,305,308,317,320,330,338,353,357,362,378
%N A243189 Nonnegative numbers of the form 2x^2 + 6xy - 3y^2.
%C A243189 Discriminant 60.
%C A243189 Nonnegative numbers of the form 5x^2 - 3y^2. - _Jon E. Schoenfield_, Jun 03 2022
%C A243189 From _Klaus Purath_, Jul 26 2023: (Start)
%C A243189 Nonnegative integers k such that 3x^2 - 5y^2 + k = 0 has integer solutions.
%C A243189 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.
%C A243189 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.
%C A243189 There are no squares in this sequence. Even powers of terms as well as products of an even number of terms belong to A243188.
%C A243189 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.
%C A243189 (End)
%H A243189 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%t A243189 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]]
%Y A243189 Primes: A141303. Cf. A243188, A107152, A237606, A141302, A243190, A141304.
%K A243189 nonn
%O A243189 1,2
%A A243189 _N. J. A. Sloane_, Jun 05 2014
%E A243189 0 prepended and more terms from _Colin Barker_, Apr 07 2015