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A068397
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Lucas(n) + (-1)^n + 1.
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2
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1, 5, 4, 9, 11, 20, 29, 49, 76, 125, 199, 324, 521, 845, 1364, 2209, 3571, 5780, 9349, 15129, 24476, 39605, 64079, 103684, 167761, 271445, 439204, 710649, 1149851, 1860500, 3010349, 4870849, 7881196, 12752045, 20633239, 33385284, 54018521, 87403805
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of domino tilings of a 2 X n strip on a cylinder.
Number of domino tilings of a 2 X n rectangle = Fibonacci(n) - see A000045.
Apart from initial terms, identical to A102081. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jun 03 2006
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REFERENCES
| S.-M. Belcastro, Tilings of 2 x n Grids on Surfaces, preprint.
H. Hosoya and F. Harary, On the matching properties of three fence graphs. J. Math. Chem., 12(1993), 211-218.
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FORMULA
| a(n) = F_(n+1) + F_(n-1) + 2 if n is even, a(n) = F_(n+1) + F_(n-1) if n is odd, where F(n) is the n-th Fibonacci number - sequence A000045.
a(n) = 1+(-1)^n+((1+sqrt(5))/2)^n+((1-sqrt(5))/2)^n = 1+(-1)^n+A000032(n). Recurrence: a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-4). G.f.: (4-3*x-4*x^2+x^3)/(1-x-2*x^2+x^3+x^4). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 08 2002
((1 + Sqrt[5])/2)^n + ((1 - Sqrt[5])/2)^n + 1 + (-1)^n (from Hosoya/Harary)
a(1) = 1, a(2) = 5; a(n) = a(n - 1) + a(n - 2) - 2 Mod[n, 2]. (from Belcastro)
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MATHEMATICA
| Table[LucasL[n] + (-1)^n + 1, {n, 1, 38}] (* From Jean-François Alcover, Sep 01 2011 *)
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CROSSREFS
| Cf. A000032, A000045.
Sequence in context: A054508 A110617 A102081 * A022344 A046588 A086654
Adjacent sequences: A068394 A068395 A068396 * A068398 A068399 A068400
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KEYWORD
| nonn,easy
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AUTHOR
| Sharon Sela (sharonsela(AT)hotmail.com), Mar 30 2002
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 08 2002
Two initial terms added, third comment amended to be consonant with new initial terms, offset changed to be consonant with initial terms, two references added, two formulae added. - Sarah-Marie Belcastro, Jul 04 2009
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