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A056918 a(n) = 9*a(n-1)-a(n-2); a(0)=2, a(1)=9. 6
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

P. Bala, Some simple continued fraction expansions for an infinite product, Part 1

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 recurrences a(n) = k*a(n - 1) +/- a(n - 2)

Index entries for sequences related to Chebyshev polynomials.

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 September 22 00:25 EDT 2017. Contains 292326 sequences.