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A305317
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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).
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0
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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