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The number of (x, y) pairs in the rows are: 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, ...

For details on the general proper representations of a negative integer k by forms with discriminant Disc = 60 = 4*15 see A378710, with references. For the Pell case x^2 - 15*y^2 only a subset of these k values is permitted, namely those that have representative reduced primitive forms (rpapfs) Fpa(-k, j) (see A378710) equivalent to the principal reduced form CR(1) = [1, 6, -6], which is in turn equivalent to the Pell form FPell = [1, 0, -15].

Some rules for the represented -A378710(n) values are: the negative of the prime factors 2, 3 and 5 of 15 are not represented, they are equivalent to some of the other three 2-cycles forms. Powers of these three primes can never occur because they cannot be lifted (see the Apostol reference, Theorem 5.20, pp. 121-122). The products -2*3 and -3*5 are represented but not -2*5 (the rpapf [-10, 10, -1] is equivalent to [-1, 6, 6], a member of the 2-cycle called CRhat in A378710). -2*3*5 is also not represented ([-30, 30, -7] is equivalent to [2, 6, -3] from the cycle Chat).

The reservoir for the odd primes >= 7 is given by Legendre(15, p) = +1 (A097956 or A038887(n), n >= 4). These primes can be lifted uniquely, but one has to find out for each case, also for the product of powers of these primes (together with 2, 3, and 5 factors) if the rpapfs reach the fundamental cycle CR.

The number of infinite families of proper solutions for k = -A378710(n) with positive y is determined by 2^P(n), where P(n) is the number of primes >= 7 in A378710(n). These numbers 2^P(n) are given in the table below, and in the first comment.

The proper family of solutions {(x(n,i), y(n,i))}_{i = -infinity ... +infinity} are found from the rpapf(-A378710(n), j) = [-A378710(n), 2*j, (15 - j^2)/A378710(n)] with the help of the formula (x(n, i), y(n, i))^T = (+ or - B15)*(-Auto15)^i*Rtvalues(n,j)^(-1)*(1, 0)^T, for the solutions of j^2 - 15 == 0 (mod(A378710(n)), for j from 0, 1,.., A378710(n) - 1, (T for transpose) where B15 = R(0)*R(3) = -Matrix([1, 3], [0, 1]), Auto15 = R(-1)*R(6) = - Matrix([1, 6], [1, 7]). For the R(t)-transformation matrix see A378710(n). Rtvalues(n,j) is the product of R(t) matrices with the t-values leading from the rpapf(-A378710(n), j) to the form CR(1). The sign of B15 is chosen such that no negative values for y appear.

The powers (- Auto15)^i = Matrix([S(i, 8) - 7*S(i-1, 8), 6*S(i-1, 8)], [S(-i, 8), S(i, 8) - S(i-1, 8)]), with the Chebyshev polynomial S(i, 8) given, for i >= -1, in A001090(i) and S_(-i, 8) = -S(i-2, 8), for i >= 2.

From Klaus Purath, Feb 17 2024: (Start)

Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.

Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].

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 forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].

If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n].

Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) = k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.

Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.

Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.

Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)