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%I #100 Apr 06 2023 08:28:53
%S 0,1,9,82,747,6805,61992,564733,5144589,46866034,426938895,3889316089,
%T 35430783696,322766369353,2940328107873,26785719340210,
%U 244011802169763,2222891938868077,20250039251982456,184473245206710181,1680509246112374085,15309056460218076946
%N Expansion of g.f.: x/(1 - 9*x - x^2).
%C For more information about this type of recurrence follow the Khovanova link and see A054413, A086902 and A178765. - _Johannes W. Meijer_, Jun 12 2010
%C 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
%C 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
%C From _Michael A. Allen_, Mar 10 2023: (Start)
%C Also called the 9-metallonacci sequence; the g.f. 1/(1-k*x-x^2) gives the k-metallonacci sequence.
%C 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)
%H G. C. Greubel, <a href="/A099371/b099371.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael A. Allen and Kenneth Edwards, <a href="https://www.fq.math.ca/Papers1/60-5/allen.pdf">Fence tiling derived identities involving the metallonacci numbers squared or cubed</a>, Fib. Q. 60:5 (2022) 5-17.
%H J. H. Han and M. D. Hirschhorn, <a href="http://www.jstor.org/stable/27642956">Another Look at an Amazing Identity of Ramanujan</a>, Mathematics Magazine, Vol. 79 (2006), pp. 302-304. See equation 6 on page 303.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Kai Wang, <a href="https://www.researchgate.net/publication/339487198_On_k-Fibonacci_Sequences_And_Infinite_Series_List_of_Results_and_Examples">On k-Fibonacci Sequences And Infinite Series List of Results and Examples</a>, 2020.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (9,1).
%F G.f.: x/(1 - 9*x - x^2).
%F a(n) = 9*a(n-1) + a(n-2), n >= 2, a(0)=0, a(1)=1.
%F 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.
%F a(n) = (ap^n - am^p)/(ap-am) with ap:= (9+sqrt(85))/2 and am:= (9-sqrt(85))/2 = -1/ap (Binet form).
%F a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1-k, k)*9^(n-1-2*k) n >= 1.
%F a(n) = F(n, 9), the n-th Fibonacci polynomial evaluated at x=9. - _T. D. Noe_, Jan 19 2006
%F 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
%F a(p) == 85^((p-1)/2)) (mod p) for odd primes p. - _Gary W. Adamson_, Feb 22 2009
%F From _Johannes W. Meijer_, Jun 12 2010: (Start)
%F a(2n+2) = 9*A097839(n), a(2n+1) = A097841(n).
%F a(3n+1) = A041151(5n), a(3n+2) = A041151(5n+3), a(3n+3) = 2*A041151(5n+4).
%F Limit_{k -> infinity} (a(n+k)/a(k)) = (A087798(n) + A099371(n)*sqrt(85))/2.
%F Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)
%F a(n) ~ 1/sqrt(85)*((9+sqrt(85))/2)^n. - _Jean-François Alcover_, Dec 04 2013
%F a(n) = [1,0] (M^n) [0,1]^T where M is the matrix [9,1; 1,0]. - _Robert Israel_, Feb 01 2015
%F E.g.f.: 2*exp(9*x/2)*sinh(sqrt(85)*x/2)/sqrt(85). - _Stefano Spezia_, Apr 06 2023
%p F:= gfun:-rectoproc({a(n)=9*a(n-1)+a(n-2),a(0)=0,a(1)=1},a(n),remember):
%p seq(F(n),n=0..30); # _Robert Israel_, Feb 01 2015
%t CoefficientList[Series[x/(1-9*x-x^2), {x,0,30}], x] (* _G. C. Greubel_, Apr 16 2017 *)
%t LinearRecurrence[{9,1}, {0,1}, 30] (* _G. C. Greubel_, Jan 24 2018 *)
%o (Sage)
%o from sage.combinat.sloane_functions import recur_gen3
%o it = recur_gen3(0,1,9,9,1,0)
%o [next(it) for i in range(1,22)] # _Zerinvary Lajos_, Jul 09 2008
%o (Sage) [lucas_number1(n,9,-1) for n in range(0, 20)] # _Zerinvary Lajos_, Apr 26 2009
%o (PARI) my(x='x+O('x^30)); concat([0], Vec(1/(1-9*x-x^2)) ) \\ _Charles R Greathouse IV_, Feb 03 2014
%o (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
%Y Row n=9 of A073133, A172236 and A352361.
%Y Cf. A099372 (squares).
%Y Cf. A041151, A054413, A086902, A087798, A099371, A097839, A178765, A243399.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Oct 18 2004
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