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A008998 a(n) = 2*a(n-1) + a(n-3), with a(0)=1 and a(1)=2. 25
1, 2, 4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332, 119833, 264300, 582932, 1285697, 2835694, 6254320, 13794337, 30424368, 67103056, 148000449, 326425266, 719953588, 1587907625, 3502240516, 7724434620, 17036776865, 37575794246 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

A transform of A000079 under the mapping g(x)->(1/(1-x^3))g(x/(1-x^3)). - Paul Barry, Oct 20 2004

The binomial transform yields 1,3,9,..., i.e., A049220 without the leading zeros. - R. J. Mathar, May 15 2008

a(n-3) is the top left entry of the n-th power of the 3 X 3 matrix [0, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3 X 3 matrix [0, 1, 0; 0, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014

a(n) equals the number of n-length words on {0,1,2} such that 0 appears only in a run which length is a multiple of 3. - Milan Janjic, Feb 17 2015

a(n) is the number of ways to fill a 1 X n strip of tiles, using only trominos, of length 3, and squares which can be chosen to have one of two possible colors. - Michael Tulskikh, Feb 12 2020

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 452

B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.

Index entries for linear recurrences with constant coefficients, signature (2,0,1).

FORMULA

a(n) = Sum_{k=0..floor(n/3)} binomial(n-2k, k)*2^(n-3k). - Paul Barry, Oct 20 2004

O.g.f.: 1/(1-2*x-x^3). - R. J. Mathar, May 15 2008

O.g.f.: exp( Sum_{n>=1} ( (1 + sqrt(1+x))^n + (1 - sqrt(1+x))^n ) * x^n/n ). - Paul D. Hanna, Dec 21 2012

G.f.: Q(0)/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 + x^2)/( x*(4*k+4 + x^2) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 30 2013

a(n) = Sum_{k=0..n} A052980(n). - Greg Dresden, May 28 2020

MAPLE

A008998 := proc(n) option remember; if n <= 2 then 2^n else 2*procname(n-1) +procname(n-3); fi; end proc;

MATHEMATICA

LinearRecurrence[{2, 0, 1}, {1, 2, 4}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)

PROG

(MAGMA) [ n eq 1 select 1 else n eq 2 select 2 else n eq 3 select 4 else 2*Self(n-1)+Self(n-3): n in [1..40]]; // Vincenzo Librandi, Aug 21 2011

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, ((1+sqrt(1+x+x*O(x^n)))^m + (1-sqrt(1+x+x*O(x^n)))^m)*x^m/m)), n)} /* Paul D. Hanna, Dec 21 2012 */

(Sage)

def A008998_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( 1/(1-2*x-x^3) ).list()

A008998_list(40) # G. C. Greubel, Feb 14 2020

(GAP) a:=[1, 2, 4];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-3]; od; a; # G. C. Greubel, Feb 14 2020

CROSSREFS

Cf. A077852, A077926. Partial sums of A052980.

Sequence in context: A129988 A035530 A141016 * A024736 A024562 A087219

Adjacent sequences:  A008995 A008996 A008997 * A008999 A009000 A009001

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 3 11:40 EDT 2020. Contains 336198 sequences. (Running on oeis4.)