cp's OEIS Frontend

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

%I A242662 #34 Feb 11 2023 16:42:14
%S A242662 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,
%T A242662 93,98,100,106,109,113,114,116,121,128,133,137,141,144,148,149,162,
%U A242662 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
%N A242662 Nonnegative integers of the form x^2 + 4xy - 3y^2.
%C A242662 Discriminant = 28.
%C A242662 Also nonnegative integers of the form x^2 - 7y^2. - _Colin Barker_, Sep 29 2014
%C A242662 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
%C A242662 For the subsequence of numbers that are properly represented see A358946. - _Wolfdieter Lang_, Jan 18 2023
%C A242662 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
%H A242662 R. J. Mathar, <a href="/A242662/b242662.txt">Table of n, a(n) for n = 0..1184</a>
%H A242662 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 A242662 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]]
%Y A242662 Primes = A141172.
%Y A242662 Cf. A242666, A358946.
%K A242662 nonn
%O A242662 0,3
%A A242662 _N. J. A. Sloane_, May 31 2014