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Search: id:A054854
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| A054854 |
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Number of ways to tile a 4 X n region with 1 X 1 and 2 X 2 tiles. |
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12
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| 1, 1, 5, 11, 35, 93, 269, 747, 2115, 5933, 16717, 47003, 132291, 372157, 1047181, 2946251, 8289731, 23323853, 65624397, 184640891, 519507267, 1461688413, 4112616845, 11571284395, 32557042499, 91602704493, 257733967693, 725161963867
(list; graph; listen)
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OFFSET
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0,3
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LINKS
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S. Heubach, Tiling an m X n area with squares of size up to k X k (m <=5), Congressus Numerantium 140 (1999), pp. 43-64.
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FORMULA
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G.f.: (1-x)/(1-2*x-3*x^2+2*x^3). - N. J. A. Sloane (njas(AT)research.att.com) Nov 17, 2002
a(n)=a(n-1)+4a(n-2)+2*( a(n-3)+a(n-4)+...+a(0) )
a(n) = 2a(n-1)+3a(n-2)-2a(n-3) - Keith Schneider (kschneid(AT)bulldog.unca.edu), Apr 02 2006
a(n) = Term (2,2) of matrix [5,1,1; 1,1,0; 1,0,1/2]*[2,1,0; 3,0,1; -2,0,0]^n - Alois P. Heinz (heinz(AT)hs-heilbronn.de), May 18 2008
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EXAMPLE
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a(2)=5 as there is one tiling of a 4x2 region with only 1 X 1 tiles, 3 tilings with exactly one 2 X 2 tile and one tiling consisting of two 2 X 2 tiles.
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MAPLE
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A := Matrix([[5, 1, 1], [1, 1, 0], [1, 0, 1/2]]); M := Matrix([[2, 1, 0], [3, 0, 1], [ -2, 0, 0]]); A054854 := n->(A.M^n)[2, 2]; seq (A054854(n), n=0..50); - Alois P. Heinz (heinz(AT)hs-heilbronn.de), May 18 2008
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MATHEMATICA
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f[{A_, B_}] := Module[{til = A, basic = B}, {Flatten[Append[til, ListConvolve[A, B]]], AppendTo[basic, 2]}]; NumOfTilings[n_] := Nest[f, {{1, 1}, {1, 4}}, n - 2][[1]] NumOfTilings[30]
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CROSSREFS
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Cf. A054855.
Sequence in context: A077917 A055936 A164560 this_sequence A005178 A065315 A065317
Adjacent sequences: A054851 A054852 A054853 this_sequence A054855 A054856 A054857
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KEYWORD
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easy,nonn
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AUTHOR
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Silvia Heubach (silvi(AT)cine.net), Apr 21 2000
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