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A107839
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a(n) = 5*a(n-1) - 2*a(n-2); a(0)=1, a(1)=5.
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19
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
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LINKS
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FORMULA
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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
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MAPLE
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a:= n-> (<<0|1>, <-2|5>>^n)[2$2]:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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