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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). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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) = A00255(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.

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

Table of n, a(n) for n=1..40.

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

EXAMPLE

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]

...

CROSSREFS

Cf. A002559, A305310, A305311.

Sequence in context: A228363 A334289 A173395 * A049089 A028327 A006890

Adjacent sequences:  A305314 A305315 A305316 * A305318 A305319 A305320

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Jul 30 2018

STATUS

approved

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)