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Search: id:A127866
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| A127866 |
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Number of L-shaped tiles in all tilings of a 2xn board with 1x1 and L-shaped tiles (where the L-shaped tiles cover 3 squares). |
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7
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| 4, 12, 52, 172, 580, 1852, 5828, 17980, 54788, 165116, 493316, 1463036, 4312068, 12641276, 36887556, 107201532, 310427652, 896045052, 2579017732, 7403843580, 21205303300, 60604891132, 172872744964, 492233179132, 1399272374276
(list; graph; listen)
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OFFSET
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2,1
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REFERENCES
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P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, to appear in the Journal of Integer Sequences
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LINKS
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S. Heubach, Tiling with Ls and Squares.
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FORMULA
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a(n) =4 (-1)^n - (2/9)[(9-5*Sqrt(3))(1+Sqrt(3))^n + (9+5*Sqrt(3))(1-Sqrt(3))^n] - (n/3)[(1-Sqrt(3))(1+Sqrt(3))^n+ (1+Sqrt(3))(1-Sqrt(3))^n]; generating function: 4x^2/((1+x)(1-2x-2x^2)^2)
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EXAMPLE
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a(2) = 4 because the 2 X 2 board can be tiled either with 4 squares or with a single L-shaped tile (in four orientations) together with a single square tile and thus all the tilings of the 2 X 2 board contain 4 L-shaped tiles
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MATHEMATICA
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Table[Coefficient[Normal[Series[4x^2/((1 + x)(1 - 2x - 2x^2)^2), {x, 0, 20}]], x, n], {n, 0, 20}]
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CROSSREFS
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Cf. A127864, A127865, A127867, A127868, A127869, A127870, A127871, A127872.
Sequence in context: A045906 A025282 A149403 this_sequence A149404 A149405 A149406
Adjacent sequences: A127863 A127864 A127865 this_sequence A127867 A127868 A127869
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KEYWORD
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easy,nonn
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AUTHOR
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Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
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EXTENSIONS
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G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.
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