A305311 Numbers k(n) used for Markoff forms determining quadratic irrationals with purely periodic continued fractions.

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

Offset: 1

Author

Wolfdieter Lang, Jun 26 2018

Keywords

• nonn

Comments

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

Examples

			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, ... .
		

References

• 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.

Crossrefs

Cf. A002559, A305310.

Formula

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