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A018913 a(n) = 9a(n - 1) - a(n - 2); a(0) = 0, a(1) = 1. 17
0, 1, 9, 80, 711, 6319, 56160, 499121, 4435929, 39424240, 350382231, 3114015839, 27675760320, 245967827041, 2186034683049, 19428344320400, 172669064200551, 1534593233484559, 13638670037160480, 121213437100959761 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Define the sequence L(a_0,a_1) by a_{n+2} is the greatest integer such that a_{n+2}/a_{n+1}<a_{n+1}/a_n for n >= 0. This is L(1,9).

For n>=2, a(n) equals the permanent of the (n-1)X(n-1) tridiagonal matrix with 9's along the main diagonal, and i's along the superdiagonal and the subdiagonal (i is the imaginary unit) [From John M. Campbell, Jul 08 2011]

REFERENCES

D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993;.

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969).

A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=9, q=-1.

W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), lhs, m=11.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for sequences related to linear recurrences with constant coefficients

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

FORMULA

G.f.: x/(1-9*x+x^2).

a(n) = S(2*n-1, sqrt(11))/sqrt(11) = S(n-1, 9); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310. S(-1, x) := 0.

a(n)={[(9+sqrt(77))/2]^n - [(9-sqrt(77))/2]^n}/sqrt(77). G.f.(x)=x/(1-9*x+x^2). - Barry E. Williams, Aug 21 2000

a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*8^k. - Philippe Deléham, Feb 10 2012

Product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + sqrt(77)). - Peter Bala, Dec 23 2012

Product {n >= 2} (1 - 1/a(n)) = 1/18*(7 + sqrt(77)). - Peter Bala, Dec 23 2012

MATHEMATICA

Join[{a=0, b=1}, Table[c=9*b-a; a=b; b=c, {n, 60}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 20 2011*)

CoefficientList[Series[x/(1 - 9*x + x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 23 2012 *)

PROG

sage: [lucas_number1(n, 9, 1) for n in range(22)] - Zerinvary Lajos, Jun 25 2008

(MAGMA) I:=[0, 1]; [n le 2 select I[n] else 9*Self(n-1) - Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 23 2012

CROSSREFS

Cf. A000027, A001906, A001353, A004254, A001109, A004187, A001090.

Cf. A056918(n)=sqrt{77*(a(n))^2 +4}, that is, a(n)=sqrt((A056918(n)^2 - 4)/77).

Sequence in context: A171314 A081108 A176174 * A192214 A127265 A055070

Adjacent sequences:  A018910 A018911 A018912 * A018914 A018915 A018916

KEYWORD

easy,nonn

AUTHOR

R. K. Guy

EXTENSIONS

More terms from James A. Sellers, Sep 07 2000

G.f. adapted to the offset by Vincenzo Librandi, Dec 23 2012

STATUS

approved

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Last modified April 25 00:50 EDT 2014. Contains 240991 sequences.