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A018913 a(n) = 9*a(n - 1) - a(n - 2) for n>1, a(0)=0, a(1)=1. 20
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
Not to be confused with the Pisot L(1,9) sequence, which is A001019. - R. J. Mathar, Feb 13 2016
Lim_{n->oo} a(n+1)/a(n) = (9 + sqrt(77))/2 = A092290 + 1 = 8.887482... - Wolfdieter Lang, Nov 16 2023
LINKS
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 12
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), 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.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Tanya Khovanova, Recursive Sequences
Wolfdieter 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.
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). - Barry E. Williams, Aug 21 2000
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*8^k. - Philippe Deléham, Feb 10 2012
From Peter Bala, Dec 23 2012: (Start)
Product {n >= 1} (1 + 1/a(n)) = 1/7*(7 + sqrt(77)).
Product {n >= 2} (1 - 1/a(n)) = 1/18*(7 + sqrt(77)). (End)
a(n) = Sum_{k = 0..n-1} binomial(n+k, 2*k+1)*7^k = Sum_{k = 0..n-1} (-1)^(n+k+1)* binomial(n+k, 2*k+1)*11^k. - Peter Bala, Jul 17 2023
EXAMPLE
G.f. = x + 9*x^2 + 80*x^3 + 711*x^4 + 6319*x^5 + 56160*x^6 + 499121*x^7 + ...
MATHEMATICA
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
(PARI) concat(0, Vec(x/(1-9*x+x^2) + O(x^30))) \\ Michel Marcus, Sep 06 2017
CROSSREFS
Cf. A056918(n)=sqrt{77*(a(n))^2 +4}, that is, a(n)=sqrt((A056918(n)^2 - 4)/77).
Cf. A092290 + 1.
Sequence in context: A081108 A176174 A242632 * A359921 A192214 A127265
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
G.f. adapted to the offset by Vincenzo Librandi, Dec 23 2012
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)