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URL: https://oeis.org/A107839

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A107839
a(n) = 5*a(n-1) - 2*a(n-2); a(0)=1, a(1)=5.
20
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
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
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, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 117. Book's website
FORMULA
a(n) = A020698(n)-2*A020698(n-1) (n>=1).
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
E.g.f.: exp(5*x/2)*(17*cosh(sqrt(17)*x/2) + 5*sqrt(17)*sinh(sqrt(17)*x/2))/17. - Stefano Spezia, Jun 17 2025
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 *)
Table[(((5 + Sqrt[17])/2)^n - ((5 - Sqrt[17])/2)^n)/Sqrt[17], {n, 20}] // Expand (* Eric W. Weisstein, Nov 03 2024 *)
LinearRecurrence[{5, -2}, {1, 5}, 20] (* Eric W. Weisstein, Nov 03 2024 *)
CoefficientList[Series[1/(1 - 5 x + 2 x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Nov 03 2024 *)
PROG
(SageMath) [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
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Jun 12 2005
STATUS
approved