A107839 a(n) = 5*a(n-1) - 2*a(n-2); a(0)=1, a(1)=5.

1, 5, 23, 105, 479, 2185, 9967, 45465, 207391, 946025, 4315343, 19684665, 89792639, 409593865, 1868384047, 8522732505, 38876894431, 177339007145, 808941246863, 3690028220025, 16832258606399, 76781236591945, 350241665746927, 1597645855550745, 7287745946259871

Offset

0,2

Comments

a(n) = A020698(n)-2*A020698(n-1) (n>=1). Kekulé numbers for certain benzenoids.

This is the number of spanning, connected subgraphs of the "ladder graph" of n squares (ladder graph = the vertices and edges of the tiling of a 1 X n rectangle by unit squares). - David Pasino (davepasino(AT)yahoo.com), Sep 18 2007

a(n) equals the number of words of length n over {0,1,2,3,4} avoiding 01 and 02. - Milan Janjic, Dec 17 2015

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).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 117. Book's website

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

Formula

a(n) = [M^(n+1)]_1,2, 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) = (((5 + s)/2)^(n+1) - ((5 - s)/2)^(n+1))/s with s = 17^(1/2). - David Pasino (davepasino(AT)yahoo.com), Jan 09 2009

G.f.: 1/(1 - 5*x + 2*x^2). - R. J. Mathar, Apr 07 2009

Maple

a:= n-> (<<0|1>, <-2|5>>^n)[2$2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 21 2020

Mathematica

a[n_]:=(MatrixPower[{{1, 2}, {1, 4}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)

Prog

(Sage) [lucas_number1(n, 5, 2) for n in range(27)] # Zerinvary Lajos, Jun 25 2008
(Magma) I:=[1, 5]; [n le 2 select I[n] else 5*Self(n-1)-2*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015
(PARI) Vec(1/(1-5*x+2*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

Crossrefs

Cf. A020698, A055099 (inverse binomial transform).

Sequence in context: A064914 A243873 A239406 * A270530 A128732 A026894

Adjacent sequences: A107836 A107837 A107838 * A107840 A107841 A107842

Keyword

nonn,easy

Author

Emeric Deutsch, Jun 12 2005

Status

approved