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 A254757 Part of the positive proper solutions x of the Pell equation x^2 - 2*y^2 = - 7^2 based on the fundamental solution (x0, y0)= (-1, 5). 3
 17, 103, 601, 3503, 20417, 118999, 693577, 4042463, 23561201, 137324743, 800387257, 4664998799, 27189605537, 158472634423, 923646201001, 5383404571583, 31376781228497, 182877282799399, 1065886915567897, 6212444210607983 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The corresponding y solutions are given in A220414. The other part of the proper (sometimes called primitive) solutions are given in (A254758(n), A254759(n)) for n >= 1. The improper positive solutions come from 7*(x(n), y(n)) with the positive proper solutions of the Pell equation x^2 - 2*y^2 = -1 given in (A001653(n-1), A002315(n)), for n >= 0. REFERENCES T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, 1964, Theorem 109, pp. 207-208 with Theorem 104, pp. 197-198. LINKS Wolfdieter Lang, Binary Quadratic Forms (indefinite case). Index entries for linear recurrences with constant coefficients, signature (6, -1). FORMULA a(n) = rational part of z(n), where z(n) = (-1+5*sqrt(2))*(3+2*sqrt(2))^n, n >= 1. G.f.: (17 + x)/(1 - 6*x + x^2). a(n) = 6*a(n-1) - a(n-2), n >= 2, with a(0) = -1 and a(1) = 17. a(n) = 17*S(n-1, 6) + S(n-2, 6), n >= 1, with Chebyshev's S-polynomials evaluated at x = 6 (see A049310). EXAMPLE The first pairs of positive solutions of this part of the Pell equation  x^2 - 2*y^2 = - 7^2 are: [17, 13], [103, 73], [601, 425], [3503, 2477], [20417, 14437], [118999, 84145], [693577, 490433], [4042463, 2858453], [23561201, 16660285], [137324743, 97103257], ... MATHEMATICA LinearRecurrence[{6, -1}, {17, 103}, 20] (* Harvey P. Dale, Sep 01 2017 *) PROG (PARI) Vec((17 + x)/(1 - 6*x + x^2) + O(x^30)) \\ Michel Marcus, Feb 08 2015 CROSSREFS Cf. A049310, A220414, A254758, A254759, A001653, A002315. Sequence in context: A041552 A160767 A078625 * A142266 A214632 A160219 Adjacent sequences:  A254754 A254755 A254756 * A254758 A254759 A254760 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Feb 07 2015 STATUS approved

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Last modified October 15 13:38 EDT 2019. Contains 328030 sequences. (Running on oeis4.)