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 A056918 a(n) = 9*a(n-1)-a(n-2); a(0)=2, a(1)=9. 7
 2, 9, 79, 702, 6239, 55449, 492802, 4379769, 38925119, 345946302, 3074591599, 27325378089, 242853811202, 2158358922729, 19182376493359, 170483029517502, 1515164889164159, 13466000972959929, 119678843867475202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS All nonnegative integer solutions of Pell equation a(n)^2 - 77*b(n)^2 = +4 together with b(n)=A018913(n), n>=0. - Wolfdieter Lang, Aug 31 2004 Except for the first term, positive values of x (or y) satisfying x^2 - 9xy + y^2 + 77 = 0. - Colin Barker, Feb 13 2014 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242. A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424-434. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (9,-1). FORMULA a(n) = 9*S(n-1, 9) - 2*S(n-2, 9) = S(n, 9) - S(n-2, 9) = 2*T(n, 9/2), with S(n, x) := U(n, x/2) (see A049310), S(-1, x) := 0, S(-2, x) := -1. S(n-1, 9)=A018913(n). U-, resp. T-, are Chebyshev's polynomials of the second, resp. first, kind. a(n) = {9*[((9+sqrt(77))/2)^n - ((9-sqrt(77))/2)^n] - 2*[((9+sqrt(77))/2)^(n-1) - ((9-sqrt(77))/2)^(n-1)]}/sqrt(77). G.f.: (2-9*x)/(1-9*x+x^2). a(n) = ap^n + am^n, with ap := (9+sqrt(77))/2 and am := (9-sqrt(77))/2. G.f.: (2-9*x)/(1-9*x+x^2). - Philippe Deléham, Nov 03 2008 From Peter Bala, Jan 06 2013: (Start) Let F(x) = product {n = 0..inf} (1 + x^(4*n+1))/(1 + x^(4*n+3)). Let alpha = 1/2*(9 - sqrt(77)). This sequence gives the simple continued fraction expansion of 1 + F(alpha) = 2.11095 50589 89701 91909 ... = 2 + 1/(9 + 1/(79 + 1/(702 + ...))). Also F(-alpha) = 0.88873 23915 40314 47623 ... has the continued fraction representation 1 - 1/(9 - 1/(79 - 1/(702 - ...))) and the simple continued fraction expansion 1/(1 + 1/((9-2) + 1/(1 + 1/((79-2) + 1/(1 + 1/((702-2) + 1/(1 + ...))))))). F(alpha)*F(-alpha) has the simple continued fraction expansion 1/(1 + 1/((9^2-4) + 1/(1 + 1/((79^2-4) + 1/(1 + 1/((702^2-4) + 1/(1 + ...))))))). Cf. A005248. (End) MATHEMATICA a[0] = 2; a[1] = 9; a[n_] := 9a[n - 1] - a[n - 2]; Table[ a[n], {n, 0, 17}] (* Robert G. Wilson v, Jan 30 2004 *) PROG (Sage) [lucas_number2(n, 9, 1) for n in range(23)] # Zerinvary Lajos, Jun 25 2008 (Haskell) a056918 n = a056918_list !! n a056918_list = 2 : 9 :    zipWith (-) (map (* 9) \$ tail a056918_list) a056918_list -- Reinhard Zumkeller, Jan 06 2013 CROSSREFS Cf. A018913. a(n)=sqrt(77*A018913(n)^2 + 4). A005248. Sequence in context: A184894 A111196 A229211 * A194471 A215629 A221460 Adjacent sequences:  A056915 A056916 A056917 * A056919 A056920 A056921 KEYWORD easy,nonn AUTHOR Barry E. Williams, Aug 21 2000 EXTENSIONS More terms from James A. Sellers, Sep 07 2000 Chebyshev comments from Wolfdieter Lang, Oct 31 2002 STATUS approved

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Last modified July 22 19:35 EDT 2018. Contains 312918 sequences. (Running on oeis4.)