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%I #10 May 27 2024 12:22:34
%S 1,8,55,379,1891,2600,17821,64261,84680,122149,421849,837224,2183005,
%T 3950155,5738419,18883369,39331711,74157931,94070600,128629621,
%U 185381839,269583560,486268651,1847753209,2519186671,3192137000,4210906771,6000283981,8707689224,12664688905,20977322059,41089519729,85578188905,86805069128,195388310755,409067053471
%N 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.
%C 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.
%C 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].
%F 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.
%Y Cf. A002559, A305312, A324250.
%K nonn,changed
%O 1,2
%A _Wolfdieter Lang_, Jul 15 2019
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