login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). - John M. Campbell, Jul 08 2011

For n>=1, a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,8}. - Milan Janjic, Jan 25 2015

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;.

LINKS

Vincenzo Librandi, 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., 5.5 (1967), 424-434. Case a=0,b=1; p=9, q=-1.

Tanya Khovanova, Recursive Sequences

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

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (9,-1).

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

EXAMPLE

G.f. = x + 9*x^2 + 80*x^3 + 711*x^4 + 6319*x^5 + 56160*x^6 + 499121*x^7 + ...

MATHEMATICA

Join[{a=0, b=1}, Table[c=9*b-a; a=b; b=c, {n, 60}]] (* 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: A081108 A176174 A242632 * A192214 A127265 A055070

Adjacent sequences:  A018910 A018911 A018912 * A018914 A018915 A018916

KEYWORD

easy,nonn

AUTHOR

R. K. Guy

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified July 4 22:42 EDT 2015. Contains 259222 sequences.