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A099371
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Expansion of g.f.: x/(1 - 9*x - x^2).
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26
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0, 1, 9, 82, 747, 6805, 61992, 564733, 5144589, 46866034, 426938895, 3889316089, 35430783696, 322766369353, 2940328107873, 26785719340210, 244011802169763, 2222891938868077, 20250039251982456, 184473245206710181
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OFFSET
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0,3
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COMMENTS
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a(p) == 85^((p-1)/2)) mod p for odd primes p. - Gary W. Adamson, Feb 22 2009
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
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
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
Kai Wang, On k-Fibonacci Sequences And Infinite Series List of Results and Examples, 2020.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (9,1).
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FORMULA
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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
From Johannes W. Meijer, Jun 12 2010: (Start)
a(2n+2) = 9*A097839(n), a(2n+1) = A097841(n).
a(3n+1) = A041151(5n), a(3n+2) = A041151(5n+3), a(3n+3) = 2*A041151(5n+4).
Limit_{k -> infinity} (a(n+k)/a(k)) = (A087798(n) + A099371(n)*sqrt(85))/2.
Lim_{n->infinity} A087798(n)/A099371(n) = sqrt(85). (End)
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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A099372 (squares), A099371.
Cf. A243399.
Sequence in context: A288789 A033119 A033127 * A334611 A068109 A163460
Adjacent sequences: A099368 A099369 A099370 * A099372 A099373 A099374
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Oct 18 2004
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STATUS
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approved
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