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 A097766 Pell equation solutions (11*a(n))^2 - 122*b(n)^2 = -1 with b(n):=A097767(n), n >= 0. 4
 1, 487, 236681, 115026479, 55902632113, 27168564180439, 13203866289061241, 6417051847919582687, 3118673994222628124641, 1515669144140349348992839, 736612085378215560982395113, 357991957824668622288095032079, 173983354890703572216453203195281 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..372 Tanya Khovanova, Recursive Sequences Giovanni Lucca, Integer Sequences and Circle Chains Inside a Hyperbola, Forum Geometricorum (2019) Vol. 19, 11-16. Index entries for linear recurrences with constant coefficients, signature (486,-1). FORMULA G.f.: (1 + x)/(1 - 2*243*x + x^2). a(n) = S(n, 2*243) + S(n-1, 2*243) = S(2*n, 2*sqrt(122)), with Chebyshev polynomials of the second kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x). a(n) = ((-1)^n)*T(2*n+1, 11*i)/(11*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120. a(n) = 486*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=487. - Philippe Deléham, Nov 18 2008 a(n) = (1/11)*sinh((2*n + 1)*arcsinh(11)). - Bruno Berselli, Apr 03 2018 EXAMPLE (x,y) = (11*1=11;1), (5357=11*487;485), (2603491=11*236681;235709), ... give the positive integer solutions to x^2 - 122*y^2 =-1. MATHEMATICA LinearRecurrence[{486, -1}, {1, 487}, 12] (* Ray Chandler, Aug 12 2015 *) PROG (PARI) x='x+O('x^99); Vec((1+x)/(1-2*243*x+x^2)) \\ Altug Alkan, Apr 05 2018 CROSSREFS Cf. A097765 for S(n, 2*243). Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775. Sequence in context: A236424 A235839 A203873 * A214806 A126819 A045011 Adjacent sequences:  A097763 A097764 A097765 * A097767 A097768 A097769 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified September 21 11:46 EDT 2020. Contains 337268 sequences. (Running on oeis4.)