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A107839
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a(n)=5a(n-1)-2a(n-2); a(0)=1, a(1)=5.
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7
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)=A020698(n)-2*A020698(n-1) (n>=1). Kekule 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
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REFERENCES
| S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 78).
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FORMULA
| a(k) = [M^k]_1,2, where M is the 3 by 3 matrix defined as follows: M = [2,1,2;1,1,1;2,1,2]. - Simone Severini (simoseve(AT)gmail.com), Jun 12 2006
a(n) = (((5 + s)/2)^(n+1) - ((5 - s)/2)^(n+1))/s with s = 17^(1/2) [From David Pasino (davepasino(AT)yahoo.com), Jan 09 2009]
G.f.: 1/(1-5*x+2*x^2). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 07 2009]
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MATHEMATICA
| a[n_]:=(MatrixPower[{{1, 2}, {1, 4}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 19 2010]
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PROG
| sage: [lucas_number1(n, 5, 2) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(Other) sage: [lucas_number1(n, 5, 2) for n in xrange(1, 24)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
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CROSSREFS
| Cf. A020698.
Sequence in context: A167660 A026760 A064914 * A128732 A026894 A126473
Adjacent sequences: A107836 A107837 A107838 * A107840 A107841 A107842
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 12 2005
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