A308687 a(n) = A305312(n)/4 if A305312(n)is even and a(n) = (A305312(n) - 1)/4 if A305312(n) is odd, for n >= 1.

1, 8, 55, 379, 1891, 2600, 17821, 64261, 84680, 122149, 421849, 837224, 2183005, 3950155, 5738419, 18883369, 39331711, 74157931, 94070600, 128629621, 185381839, 269583560, 486268651, 1847753209, 2519186671, 3192137000, 4210906771, 6000283981, 8707689224, 12664688905, 20977322059, 41089519729, 85578188905, 86805069128, 195388310755, 409067053471

Offset: 1

Author

Wolfdieter Lang, Jul 15 2019

Keywords

• nonn

Comments

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].

Crossrefs

Cf. A002559, A305312, A324250.

Formula

a(n) = A305312(n)/4 or a(n) = (A305312(n) - 1)/4 if A305312(n) is even or odd, respectively, where A305312(n) = Disc(n) = b(n)*(b(n) + 4) with b(n) = 3*m(n) - 2 = A324250(n), and m(n) = A002559(n), for n >= 1.