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A008998 a(n) = 2*a(n-1) + a(n-3). 20
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 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 3X3 matrix [0, 0, 1; 1, 1, 1; 0, 1, 1] or of the 3X3 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

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/(1-2x-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

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}, 80] (* 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 */

CROSSREFS

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 June 19 13:11 EDT 2019. Contains 324222 sequences. (Running on oeis4.)