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A305311
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Numbers k(n) used for Markoff forms determining quadratic irrationals with purely periodic continued fractions.
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2
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2, 5, 12, 31, 70, 81, 212, 408, 463, 555, 1045, 1453, 2378, 3157, 3804, 6914, 9959, 13860, 15605, 18045, 21622, 26073, 35491, 68260, 80782, 90903, 103247, 123042, 148183, 178707, 233030, 321983, 470832, 467861, 703292, 1015645, 1205641, 1224876, 1541791, 2205232
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange Spectra, Am. Math. Soc., Providence. Rhode Island, 1989.
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LINKS
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FORMULA
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a(n) = A305310(n) + 2, n >= 1. The proof is based on Theorem 3, pp. 23-24, of the Cusick-Flahive reference. See also the W. Lang link under A305310. - Wolfdieter Lang, Jul 29 2018
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EXAMPLE
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The form coefficients [m(n), 3*m(n) - 2*k(n), l(n) - 3*k(n)] with l(n) := (k(n)^2 +1)/m(n), n >= 1, begin: [1, -1, -1], [2, -4, -2], [5, -9, -7], [13, -23, -19], [29, -53, -41], [34, -60, -50], [89, -157, -131], [169, -309, -239], [194, -344, -284], [233, -411, -343], [433, -791, -613], [610, -1076, -898], [985, -1801, -1393], [1325, -2339, -1949], [1597, -2817, -2351], [2897, -5137, -4241], [4181, -7375, -6155], [5741, -10497, -8119], [6466, -11812, -9154], [7561, -13407, -11069], ... .
The corresponding quadratic irrationals xi(n) with purely periodic continued fraction representations begin: (1 + sqrt(5))/2, 1 + sqrt(2), (9+sqrt(221))/10, (23 + sqrt(1517))/26, (53 + sqrt(7565))/56, (15 + 5*sqrt(26))/17, (157 + sqrt(71285))/178, (309 + sqrt(257045))/338, (86 + sqrt(21170))/97, (411 + sqrt(488597))/466, (791 + sqrt(1687397))/866, (269 + sqrt(209306))/305, (1801 + sqrt(8732021))/1970, (2339 + sqrt(15800621))/2650, (2817 + sqrt(22953677))/3194, (5137 + sqrt(75533477))/5794, (7375 + sqrt(157326845))/8362, (10497 + 5*sqrt(11865269))/11482, (2953 + 5*sqrt(940706))/3233, (13407 + sqrt(514518485))/15122, ... .
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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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