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Equivalently, positive numbers of the form k = x^2 + 2xy - 2y^2. These are equivalent forms, of discriminant 12.

Also numbers representable as x^2 + 4*x*y + y^2 with 0 <= x <= y. - Gheorghe Coserea, Jul 29 2018 [The restriction 0 <= x <= y is not necessary. - Klaus Purath, Feb 05 2023]

From Klaus Purath, Feb 05 2023: (Start)

Also positive numbers of the form x^2 + 2*m*x*y + (m^2 - 3)*y^2. This includes all forms given above so far.

All terms are congruent to {0, 1, 4, 6, 9, 10} modulo 12.

The product of any two terms belongs to the sequence - (empirically secured up to a(k)*a(m) for 2 <= k, m <= 85). Thus it appears that this sequence is closed under multiplication. Perhaps someone can find a proof? (End)

The array A(n, k) gives the number of the representative parallel binary quadratic primitive forms for discriminant Disc(n) = 4*D(n) = 4*A000037(n) and representation of positive integer k which are (properly) equivalent to the Pell form F(n) = [1, 0, -D(n)].

For the definition of representative parallel primitive forms for discriminant Disc > 0 (the indefinte case) and representation of nonzero integers k see the Scholz-Schoeneberg reference, p. 105, or the Buell reference p. 49 (without use of the name parallel). For the procedure to find the primitive representative parallel forms (rpapfs) for Disc(n) = 4*D(n) = 4*A000037(n) and nonzero integer k see the W. Lang link given in A324251, section 3.

Among them the parallel forms which are equivalent to the reduced principal form F_p(n) = [1, 2*s(n), -(D(n) - s(n))^2], with s(n) = A000194(n), are important to find all solutions (x, y) with gcd(x, y) = 1 (proper) of the Pell form F(n) = [1, 0, -D(n)] with Disc(F(n)) = 4*D(n) > 0 representing a positive integer k. The number of these parallel forms pa(n, k) gives the number of the proper fundamental solutions. The general solution is obtained from the fundamental solutions with the help of integer powers of the automorphic matrix corresponding to the cycle determined by the reduced principal form F_p(n).

Thus the array A(n,k) gives the number of proper families (also called classes) of solutions of the Pell equation x^2 - Dn(n)*y^2 = k, for positive integer k. The positions of the nonzero entries in row n give the list of the k values for which proper solutions exist.

These position lists are A057126 (conjecture) and A243655, for k = 1 and 2.

The first column has only 1s, showing that every Pell form [1, 0, -D(n)] represents k = +1, and that there is only one family of proper solutions.

Discriminant 205.

Also, nonnegative numbers represented by the indefinite quadratic form x^2-61y^2, of discriminant 244. The corresponding reduced form is x^2+14xy-12y^2.

Also 12*a(n) has the form z^2 - 61*y^2, where z = 6*x+5*y. [Bruno Berselli, Jun 20 2014]

Calculated using Dario Alpern's quadratic Diophantine solver, see link.

Discriminant 205.

Discriminant 205.

12*a(n) has the form z^2 - 205*y^2, where z = 6*x+13*y. In fact, this is a particular case of the following identity on the numbers of the form a*x^2+b*x*y+c*y^2: 4*a*(a*x^2+b*x*y+c*y^2) = (2*a*x+b*y)^2-(b^2 -4*a*c)*y^2. [Bruno Berselli, Jun 20 2014]

Discriminant of indefinite binary quadratic form: 229.

Discriminant of indefinite binary quadratic form: 148.