OFFSET
0,3
COMMENTS
For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - Johannes W. Meijer, Jun 12 2010
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 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jul 08 2011
For n >= 1, a(n) equals the number of words of length n-1 on alphabet {0,1,...,9} avoiding runs of zeros of odd lengths. - Milan Janjic, Jan 28 2015
From Michael A. Allen, Mar 10 2023: (Start)
Also called the 9-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
a(n+1) is the number of tilings of an n-board (a board with dimensions n X 1) using unit squares and dominoes (with dimensions 2 X 1) if there are 9 kinds of squares available. (End)
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael A. Allen and Kenneth Edwards, Fence tiling derived identities involving the metallonacci numbers squared or cubed, Fib. Q. 60:5 (2022) 5-17.
J. H. Han and M. D. Hirschhorn, Another Look at an Amazing Identity of Ramanujan, Mathematics Magazine, Vol. 79 (2006), pp. 302-304. See equation 6 on page 303.
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (9,1).
FORMULA
G.f.: x/(1 - 9*x - x^2).
a(n) = 9*a(n-1) + a(n-2), n >= 2, a(0)=0, a(1)=1.
a(n) = (-i)^(n-1)*S(n-1, 9*i) with S(n, x) Chebyshev's polynomials of the second kind (see A049310) and i^2=-1.
a(n) = (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2 = -1/ap (Binet form).
a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k, k)*9^(n-1-2*k) n >= 1.
a(n) = F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - T. D. Noe, Jan 19 2006
a(n) = ((9+sqrt(85))^n - (9-sqrt(85))^n)/(2^n*sqrt(85)). Offset 1. a(3)=82. - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009
a(p) == 85^((p-1)/2)) (mod p) for odd primes p. - Gary W. Adamson, Feb 22 2009
From Johannes W. Meijer, Jun 12 2010: (Start)
a(n) ~ 1/sqrt(85)*((9+sqrt(85))/2)^n. - Jean-François Alcover, Dec 04 2013
a(n) = [1,0] (M^n) [0,1]^T where M is the matrix [9,1; 1,0]. - Robert Israel, Feb 01 2015
E.g.f.: 2*exp(9*x/2)*sinh(sqrt(85)*x/2)/sqrt(85). - Stefano Spezia, Apr 06 2023
MAPLE
F:= gfun:-rectoproc({a(n)=9*a(n-1)+a(n-2), a(0)=0, a(1)=1}, a(n), remember):
seq(F(n), n=0..30); # Robert Israel, Feb 01 2015
MATHEMATICA
CoefficientList[Series[x/(1-9*x-x^2), {x, 0, 30}], x] (* G. C. Greubel, Apr 16 2017 *)
LinearRecurrence[{9, 1}, {0, 1}, 30] (* G. C. Greubel, Jan 24 2018 *)
PROG
(Sage)
from sage.combinat.sloane_functions import recur_gen3
it = recur_gen3(0, 1, 9, 9, 1, 0)
[next(it) for i in range(1, 22)] # Zerinvary Lajos, Jul 09 2008
(Sage) [lucas_number1(n, 9, -1) for n in range(0, 20)] # Zerinvary Lajos, Apr 26 2009
(PARI) my(x='x+O('x^30)); concat([0], Vec(1/(1-9*x-x^2)) ) \\ Charles R Greathouse IV, Feb 03 2014
(Magma) I:=[0, 1]; [n le 2 select I[n] else 9*Self(n-1) + Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Oct 18 2004
STATUS
approved