A305317 a(n) gives the length of the period of the regular continued fraction of the quadratic irrational of any Markoff form representative Mf(n), n >= 1 (assuming the uniqueness conjecture).

1, 1, 4, 6, 6, 8, 10, 8, 10, 12, 10, 14, 10, 14, 16, 14, 18, 12, 14, 16, 18, 20, 14, 22, 14, 16, 18, 20, 22, 24, 18, 22, 16, 26, 22, 26, 18, 28, 22, 26

Offset: 1

Author

Wolfdieter Lang, Jul 30 2018

Keywords

• nonn

Comments

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.

Examples

			The periods for the representative form Mf(n) with k(n) = A305311(n) are given for n=1..40 in the W. Lang link in Table 2.
The first 11 examples (given by Perron) are:
  n     periods             length  quadratic irrationals xi  Markoff form coeffs.
  1:    (1)                    1    (1 + sqrt(5))/2           [1, -1, -1]
  2:    (2)                    1     1 + sqrt(2)              [2, -4, -2]
  3:    (2_2, 1_2)             4    (9 + sqrt(221))/10        [5, -9, -7]
  4:    (2_2, 1_4)             6    (23 + sqrt(1517))/26      [13, -23,-19]
  5:    (2_4, 1_2)             6    (53 + sqrt(7565))/58      [29, -53, -4]
  6:    (2_2, 1_6)             8    (15 + 5*sqrt(26))/17      [34, -60, -50]
  7:    (2_2, 1_8)            10    (157 + sqrt(71285))/178   [89, -157, -131]
  8:    (2_6, 1_2)             8    (309 + sqrt(257045))/338  [169, -309, -239]
  9:    (2_2, 1_2, 2_2, 1_4)  10    (86 + sqrt(21170))/97     [194, -344, -284]
  10:   (2_2, 1_10)           12    (411 + sqrt(488597))/466  [233, -411, -343]
  11:   (2_4, 1_2, 2_2, 1_2)  10    (791 + sqrt(1687397))/866 [433, -791, -613]
  ...
		

References

• Aigner, Martin. Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013.
• Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, II, pp. 1-12, Sitzungsber. Heidelberger Akademie der Wiss., 1921, 8. Abhandlung, Carl Winters Universitätsbuchhandlung.

Links

• Wolfdieter Lang, A Note on Markoff Forms Determining Quadratic Irrationals with Purely Periodic Continued Fractions

Crossrefs

Cf. A002559, A305310, A305311.