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A008998 a(n) = 2*a(n-1) + a(n-3), with a(0)=1 and a(1)=2. 26
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
B. Rittaud, On the Average Growth of Random Fibonacci Sequences, Journal of Integer Sequences, 10 (2007), Article 07.2.4.
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: A035530 A350325 A141016 * A024736 A024562 A087219
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)