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For the definition of representative parallel primitive forms (rpapfs) for discriminant Disc > 0 (the indefinite 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.

Note that the number of rpapfs of a discriminant Disc > 0 for k >= 1 is identical with the one for negative k. These forms differ in the signs of the a and c entries of these forms but not the b >= 0 entry (called an outer sign flip). See some examples below, and the program in the mentioned W. Lang link, section 3.

For the forms counted in the array A(n, k) see Table 3 of the W. Lang link given in A324251, for n = 1..30 and k = 1..10.

Compare the present array with the ones given in A324252 and A307303 for the number of rpapfs for discriminant 4*D(n) and representable positive and negative k, respectively, that are equivalent (under SL(2, Z)) to the reduced principal form F_p = [1, 2*s(n), -(D(n) - s(n)^2)] with s(n) = A000194(n), of the unreduced Pell form F(n) = [1, 0, -D(n)].

The rpapfs not counted in A324252 and A307303 are equivalent to forms of non-principal cycles for discriminant 4*D(n).

The total number of cycles (the class number h(n)) for discriminant 4*D(n) is given in A307359(n).

The array for the length of the periods of these cycles is given in A307378.

One half of the sum of the length of the periods is given in A307236.

The length of row n is 2*A307236(n). This is the number of primitive reduced binary quadratic forms of discriminant 4*D(n), with D(n) = A000037(n).

The number of cycles in row n is A307359(n), the class number h(n) of binary quadratic forms of discriminant 4*D(n).

The principal cycle starts with F_p(n) = [1, 2*s(n), -(D(n) -s(n))^2], with s(n) = A000194(n). Its period is A307372(n). This is the only cycle (the class number is 1) for n = 1, 3, 10, 13, 24, ...

For class number h(n) >= 2 the cycles come mostly in pairs of cycles which can be transformed into each other by a sign flip operation on the outer entries of the forms of the cycle (called outer sign flip). Exceptions occur if cycles are identical with their outer sign flipped ones. This happens, e.g., for n = 7 with two cycles: one of length 2 (the principal cycle CR(2)) and one of length 6. This 6-cycle is also identical to the outer sign flipped one. See the example below.

See the Buell and Scholz-Schoeneberg references for cycles and class number, and also the W. Lang link given in A324251, with Table 2.

These numbers a(n), depending on the parity of the discriminants of Markoff forms Disc(n) = b(n)*(b(n) + 4) = A305312(n), with b(n) = A324250(n), enter the definition of representative parallel forms of Disc(n) and representation -m(n)^2, where m(n) = A002559(n) = (b(n) + 2)/3 are the Markoff numbers, in the following way. FPara(n) := [-m(n)^2, 2*j(n), -(j^2(n) - a(n))/m(n)^2] or [-m(n)^2, 2*j(n) + 1, -(j(n)^2 +j(n) - a(n))/m(n)^2], if Disc(n) is even or odd, respectively, with j(n) from the interval [0, m(n)^2 - 1] such that the third member of FPara(n) becomes an integer. See the W. Lang link in A324251, section 3 for representative parallel forms, and the Buell and Scholz-Schoeneberg references given there.

The trivial solution (x = 1, y = 0) of each of the #rpapfs (number of representative parallel and primitive forms) Fpara(n;k), for k = 1, 2, ..., #rpapfs, representing -m(n)^2 leads to a fundamental solution of any primitive form F = [a, b, c] = a*x^2 + b*x*y + c*y^2 of discriminant Disc := b^2 - 4*a*c and representing - m(n)^2, by a certain proper (determinant +1) equivalence transformation. For the Markoff triples the principal reduced form F_p = [1, b(n), -b(n)], representing -m(n)^2 is of interest. It is a member of a 2-cycle of reduced forms together with F = [-b(n), b(n), 1].