A052984 a(n) = 5*a(n-1) - 2*a(n-2) for n>1, with a(0) = 1, a(1) = 3.

1, 3, 13, 59, 269, 1227, 5597, 25531, 116461, 531243, 2423293, 11053979, 50423309, 230008587, 1049196317, 4785964411, 21831429421, 99585218283, 454263232573, 2072145726299, 9452202166349, 43116719379147, 196679192563037

Offset

0,2

Comments

a(n) = A020698(n) - 4*A020698(n-1) + 4*A020698(n-2) (n>=2). Kekulé numbers for certain benzenoids. - Emeric Deutsch, Jun 12 2005

References

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).

Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Links

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1058

Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. See p. 3.

Zeying Xu, Graphical zonotopes with the same face vector, arXiv:1809.08764 [math.CO], 2018.

Index entries for linear recurrences with constant coefficients, signature (5,-2).

Formula

a(n) = A005824(2n).

G.f.: (1-2*x)/(1-5*x+2*x^2).

a(n) = Sum_{alpha=RootOf(1-5*z+2*z^2)} (1 + 6*alpha)*alpha^(-1-n)/17.

a(n) = [M^(n+1)]_2,2, where M is the 3 X 3 matrix defined as follows: M = [2,1,2; 1,1,1; 2,1,2]. - Simone Severini, Jun 12 2006

a(n-1) = Sum_{k=0..n} A147703(n,k)*(-1)^k*2^(n-k), n>1. - Philippe Deléham, Nov 29 2008

a(n) = (a(n-1)^2 + 2^n)/a(n-2). - Irene Sermon, Oct 29 2013

a(n) = A107839(n) - 2*A107839(n-1). - R. J. Mathar, Feb 27 2019

Maple

spec:= [S, {S=Sequence(Union(Prod(Sequence(Union(Z, Z)), Union(Z, Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
a[0]:=1: a[1]:=3: for n from 2 to 25 do a[n]:=5*a[n-1]-2*a[n-2] od: seq(a[n], n=0..25); # Emeric Deutsch

Mathematica

a[0]=1; a[1]=3; a[n_]:= a[n] = 5a[n-1]-2a[n-2]; Table[ a[n], {n, 0, 30}]
LinearRecurrence[{5, -2}, {1, 3}, 30] (* Harvey P. Dale, Apr 08 2014 *)
CoefficientList[Series[(1-2x)/(1-5x+2x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 09 2014 *)

Prog

(PARI) Vec((1-2*x)/(1-5*x+2*x^2)+O(x^30)) \\ Charles R Greathouse IV, Nov 20 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-2*x)/(1-5*x+2*x^2) )); // G. C. Greubel, Feb 10 2019
(Magma) a:=[1, 3]; [n le 2 select a[n] else 5*Self(n-1)-2*Self(n-2):n in [1..25]]; // Marius A. Burtea, Oct 23 2019
(Sage)
def A052984_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1-2*x)/(1-5*x+2*x^2) ).list()
A052984_list(30) # G. C. Greubel, Feb 10 2019
(GAP) a:=[1, 3];; for n in [3..30] do a[n]:=5*a[n-1]-2*a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

Crossrefs

Cf. A005824, A020698.

Sequence in context: A268596 A199297 A152594 * A360143 A262664 A151229

Adjacent sequences: A052981 A052982 A052983 * A052985 A052986 A052987

Keyword

nonn,easy

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Extensions

Edited by Robert G. Wilson v, Dec 29 2002

Status

approved