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A107839 a(n) = 5a(n-1) - 2a(n-2); a(0)=1, a(1)=5. 11

%I

%S 1,5,23,105,479,2185,9967,45465,207391,946025,4315343,19684665,

%T 89792639,409593865,1868384047,8522732505,38876894431,177339007145,

%U 808941246863,3690028220025,16832258606399,76781236591945,350241665746927

%N a(n) = 5a(n-1) - 2a(n-2); a(0)=1, a(1)=5.

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

%C 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

%C 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

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

%H Tomislav Doslic, <a href="http://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607. See Cor. 3.7(e).

%H A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, <a href="http://dx.doi.org/10.1007/978-3-0348-0237-6">The Tower of Hanoi - Myths and Maths</a>, Birkhäuser 2013. See page 117. <a href="http://tohbook.info">Book's website</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-2).

%F a(k) = [M^k]_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

%F 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

%F G.f.: 1/(1 - 5*x + 2*x^2). - _R. J. Mathar_, Apr 07 2009

%t 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 *)

%o (Sage) [lucas_number1(n,5,2) for n in range(27)] # _Zerinvary Lajos_, Jun 25 2008

%o (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

%o (PARI) Vec(1/(1-5*x+2*x^2) + O(x^100)) \\ _Altug Alkan_, Dec 17 2015

%Y Cf. A020698, A055099 (inverse binomial transform).

%K nonn,easy

%O 0,2

%A _Emeric Deutsch_, Jun 12 2005

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Last modified July 13 13:48 EDT 2020. Contains 335688 sequences. (Running on oeis4.)