cp's OEIS Frontend

Subsequence of A079896.

For the Markoff form f_{m(n)}(x, y) = m(n)*F_{m(n)}(x, y) of Cassels (pp. 31-39), see the comments on A305310. Some references are given in A002559, A305308 and A305310.

f_m(x, y) is an indefinite binary quadratic form because the discriminant is positive.

a(n) is also the discriminant D(n) = a(n) of the indefinite binary quadratic form determining the Markoff triple MT(n) = (x(n), y(n), m(n)) if the largest member is m(n) = A002559(n) and x(n) <= y(n) <= m(n). This is the form x^2 - 3*m*x*y + y^2 = -m^2 (with dropped argument n), or in reduced version X^2 + b*X*Y - b*Y^2 = -m^2, with b = b(n) = 3*m(n) - 2, where X = X(n) = y(n) - x(n) and Y = Y(n) = y(n). The uniqueness of such Markoff triples MT(n) with given largest members m(n) is a conjecture.

To find reduced forms one needs f(n) := ceiling(sqrt(D(n))) which is 3*m(n) because (3*m-1)^2 < 9*m^2 - 4 < (3*m)^2, due to 6*m(n) > 5, for n >= 1.

If the forms for a Markoff triple with largest member m are numerated with n giving m as m(n) = A002559(n)as in the present entry then the uniqueness conjecture is assumed to be true. Otherwise certain m(n) will lead to several different forms. - Wolfdieter Lang, Jul 30 2018

The indefinite binary quadratic Markoff form MF(n) = f_{m(n}(x, y) = m(n)*x^2 +(3*m(n) - 2*k(n))*x*y + ((k(n)^2 +1)/m(n) - 3*k(n))*y^2 with m(n) = A002559(n), for n >= 1, leads to purely periodic continued fractions for the solution x = xi(n) of f_{m(n)}(x, 1) = 0 with positive square root, namely xi(n) = ((2*k(n) - 3*m(n)) + sqrt(D(n)))/(2*m(n)) with discriminant D(n) = A305312(n). This form f_{m(n)}(x, y) is equivalent to the form fC_{m(n)}(x, y) given by Cassels (p. 31) with the k-sequence given in A305310.

The uniqueness conjecture (see A305310, also for the Aigner reference) is here assumed to be true. - Wolfdieter Lang, Jul 29 2018

See the Aigner reference, Corollary 3.17., p. 58. If this congruence is solvable uniquely for integer x in the given interval then the Markoff uniqueness conjecture is true.

For the values k(n) = (a(n)^2 + 1)/m(n), for n >= 3, see A309161.

Many of these values coincide with A305310.

The index n enumerates the Markoff triples with largest member m from A002559 in increasing order. If the Markoff-Frobenius uniqueness conjecture (see, e.g. the book of Aigner) is true then the triples can be numbered by n if the largest member is m(n) = A002559(n). In the other (unlikely) case there may be more than one triple (hence forms) for some Markoff numbers m from A002559, and then one orders these triples lexicographically.

The indefinite binary quadratic Markoff form Mf(n) = Mf(n;x,y) for the given Markoff number m(n) = A002559(n), n >= 1, (assuming that the mentioned uniqueness conjecture is true) is m(n)*x^2 + (3*m(n) - 2*k(n))*x*y + (l(n) - 3*k(n))y^2 with l(n) = (k(n)^2 +1)/m(n), and k(n) is defined for the representative form (of the unimodualar equvivalence class), e.g., in Cassels as k(n) = k_C(n) = A305310(n). The qudadratic irrational xi(n) is the solution of Mf(n;x,1) = 0 with the positive root. For the representative forms used by Cassels the regular continued fractions for xi(n) = xi_C(n) are not purely periodic. The smallest preperiod is -1 for n = 1 and 0 for n >= 2.

For the representative Mf(n) with k(n) = A305311(n) = k_C(n) + 2*m(n) one obtains purely periodic regular continued fractions for the quadratic irrationals xi(n). They were considered by Perron, pp. 5-6, for n=1..11. See the examples below, and in the W. Lang link, Table 2.