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 A097775 Pell equation solutions (14*a(n))^2 - 197*b(n)^2 = -1 with b(n) = A097776(n), n>=0. 16
 1, 787, 618581, 486203879, 382155630313, 300373839222139, 236093455472970941, 185569155627915937487, 145857120230086453893841, 114643510931692324844621539, 90109653735189937241418635813, 70826073192348358979430203127479, 55669203419532074967894898239562681 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Colin Barker, Table of n, a(n) for n = 0..345 Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (786,-1). FORMULA G.f.: (1 + x)/(1 - 2*393*x + x^2). a(n) = S(n, 2*393) + S(n-1, 2*393) = S(2*n, 2*sqrt(197)), with Chebyshev polynomials of the 2nd 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, 14*I)/(14*I) with the imaginary unit I and Chebyshev polynomials of the first kind. See the T-triangle A053120. a(n) = 786*a(n-1)-a(n-2) for n>1; a(0)=1, a(1)=787. - Philippe Deléham, Nov 18 2008 a(n) = (1/14)*sinh((2*n + 1)*arcsinh(14)). - Bruno Berselli, Apr 05 2018 EXAMPLE (x,y) = (14*1=14;1), (11018=14*787;785), (8660134=14*618581;617009), ... give the positive integer solutions to x^2 - 197*y^2 =-1. MATHEMATICA LinearRecurrence[{786, -1}, {1, 787}, 20] (* Harvey P. Dale, Dec 12 2017 *) PROG (PARI) Vec((1+x)/(1-2*393*x+x^2) + O(x^100)) \\ Colin Barker, Apr 04 2015 CROSSREFS Cf. A097774 for S(n, 2*393). Cf. similar sequences of the type (1/k)*sinh((2*n + 1)*arcsinh(k)): A002315 (k=1), A049629 (k=2), A097314 (k=3), A078989 (k=4), A097726 (k=5), A097729 (k=6), A097732 (k=7), A097735 (k=8), A097738 (k=9), A097741 (k=10), A097766 (k=11), A097769 (k=12), A097772 (k=13), this sequence (k=14). Sequence in context: A045231 A267476 A068660 * A072730 A180100 A072722 Adjacent sequences:  A097772 A097773 A097774 * A097776 A097777 A097778 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 31 2004 STATUS approved

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Last modified October 22 17:03 EDT 2018. Contains 316490 sequences. (Running on oeis4.)