OFFSET
0,2
COMMENTS
Chebyshev or generalized Fibonacci sequence.
This is the m=13 member of the m-family of sequences S(n,m-2) = S(2*n+1,sqrt(m))/sqrt(m). The m=4..12 (nonnegative) sequences are A000027, A001906, A001353, A004254, A001109, A004187, A001090, A018913 and A004189. The m=1..3 (signed) sequences are A049347, A056594, A010892.
All positive integer solutions of Pell equation b(n)^2 - 117*a(n)^2 = +4 together with b(n+1)=A057076(n+1), n >= 0. - Wolfdieter Lang, Aug 31 2004
For positive n, a(n) equals the permanent of the tridiagonal matrix of order n with 11'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
a(n) equals the number of 01-avoiding words of length n-1 on alphabet {0,1,...,10}. - Milan Janjic, Jan 25 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..900
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.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
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
S. Falcon, Generalized Fibonacci Sequences Generated from a k-Fibonacci Sequence, Journal of Mathematics Research Vol. 4, No. 2; April 2012. - From N. J. A. Sloane, Sep 22 2012
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
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=11, 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=13.
Index entries for linear recurrences with constant coefficients, signature (11,-1).
FORMULA
Recursion: a(n) = 11*a(n-1) - a(n-2), n >= 1; a(-1)=0, a(0)=1.
a(n) = S(2*n+1, sqrt(13))/sqrt(13) = S(n, 11); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind, A049310.
G.f.: 1/(1 - 11*x + x^2).
a(n) = ((11+3*sqrt(13))^(n+1) - (11-3*sqrt(13))^(n+1))/(2^(n+1)*3*sqrt(13)). - Rolf Pleisch, May 22 2011
a(n) = Sum_{k=0..n} A101950(n,k)*10^k. - Philippe Deléham, Feb 10 2012
From Peter Bala, Dec 23 2012: (Start)
Product_{n>=0} (1 + 1/a(n)) = 1/3*(3 + sqrt(13)).
Product_{n>=1} (1 - 1/a(n)) = 3/22*(3 + sqrt(13)). (End)
a(n) = sqrt((A057076(n+1)^2 - 4)/117).
a(n) = -a(-2-n) for all n in Z. - Michael Somos, Jul 14 2018
E.g.f.: exp(11*x/2)*(39*cosh(3*sqrt(13)*x/2) + 11*sqrt(13)*sinh(3*sqrt(13)*x/2))/39. - Stefano Spezia, Aug 07 2024
EXAMPLE
G.f. = 1 + 11*x + 120*x^2 + 1309*x^3 + 14279*x^4 + 155760*x^5 + ...
MAPLE
with(combinat):seq(fibonacci(2*n+2, 3)/3, n=0..20); # Zerinvary Lajos, Apr 20 2008
MATHEMATICA
Join[{a=1, b=11}, Table[c=11*b-a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 20 2011 *)
CoefficientList[Series[1/(1-11*x+x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 13 2012 *)
Table[Fibonacci[2n + 2, 3]/3, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)
a[ n_] := ChebyshevU[n, 11/2]; (* Michael Somos, Jul 14 2018 *)
PROG
(Sage) [lucas_number1(n, 11, 1) for n in range(1, 20)] # Zerinvary Lajos, Jun 25 2008
(PARI) Vec(1/(1-11*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
(PARI) {a(n) = polchebyshev(n, 2, 11/2)}; /* Michael Somos, Jul 14 2018 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Wolfdieter Lang, Oct 31 2002
STATUS
approved