A127864
Number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
Original entry on oeis.org
1, 1, 5, 11, 33, 87, 241, 655, 1793, 4895, 13377, 36543, 99841, 272767, 745217, 2035967, 5562369, 15196671, 41518081, 113429503, 309895169, 846649343, 2313089025, 6319476735, 17265131521, 47169216511, 128868696065, 352075825151, 961889042433, 2627929735167
Offset: 0
Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
a(2) = 5 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.
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
- S. Heubach, Tiling with Ls and Squares, 2005.
- Index entries for linear recurrences with constant coefficients, signature (1,4,2).
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I:=[1,1,5]; [n le 3 select I[n] else Self(n-1) + 4*Self(n-2) + 2*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2022
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CoefficientList[Series[1/(1-x-4*x^2-2*x^3), {x,0,30}], x]
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A028860 = BinaryRecurrenceSequence(2,2,-1,1)
def A127864(n): return A028860(n+2) + (-1)^n
[A127864(n) for n in range(51)] # G. C. Greubel, Dec 08 2022
A127865
Number of square tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
Original entry on oeis.org
2, 8, 30, 108, 354, 1152, 3614, 11204, 34170, 103176, 308598, 916236, 2702834, 7929872, 23155182, 67333140, 195082218, 563367960, 1622185958, 4658753564, 13347741666, 38160007200, 108881256414, 310108078116, 881761288154
Offset: 1
Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
a(2) = 8 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 8 square tiles.
- P. Z. Chinn, R. Grimaldi and S. Heubach, Tiling with Ls and Squares, J. Int. Sequences 10 (2007) #07.2.8.
- S. Heubach, Tiling with Ls and Squares, 2005.
- Index entries for linear recurrences with constant coefficients, signature (2, 7, -4, -20, -16, -4).
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Table[(2n - 12)(-1)^n + (2/3)((9 - 5Sqrt[3])(1 + Sqrt[3])^n + (9 + 5Sqrt[3])(1 - Sqrt[3])^n) + (n/Sqrt[3])((Sqrt[3] - 1)( 1 + Sqrt[3])^n + (Sqrt[3] + 1)(1 - Sqrt[3])^n), {n, 1, 30}]
A127866
Number of L-shaped tiles in all tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares).
Original entry on oeis.org
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
Offset: 2
Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007
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.
- P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8
- Index entries for linear recurrences with constant coefficients, signature (3, 4, -8, -12, -4).
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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}]
G.f. proposed by Maksym Voznyy checked and corrected by
R. J. Mathar, Sep 16 2009.
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