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A054854 Number of ways to tile a 4 X n region with 1 X 1 and 2 X 2 tiles. 16
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; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

S. Heubach, Tiling an m-by-n area with squares of size up to k-by-k (m<=5), Congressus Numerantium 140 (1999), 43-64.

Index entries for linear recurrences with constant coefficients, signature (2,3,-2).

FORMULA

G.f.: (1-x)/(1-2*x-3*x^2+2*x^3). - N. J. A. Sloane, Nov 17 2002

a(n) = a(n-1)+4*a(n-2)+2*( a(n-3)+a(n-4)+...+a(0) ).

a(n) = 2*a(n-1)+3*a(n-2)-2*a(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, May 18 2008

EXAMPLE

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.

MAPLE

A:= Matrix([[5, 1, 1], [1, 1, 0], [1, 0, 1/2]]); M:= Matrix([[2, 1, 0], [3, 0, 1], [ -2, 0, 0]]): a:= n->(A.M^n)[2, 2]: seq(a(n), n=0..50); # Alois P. Heinz, May 18 2008

MATHEMATICA

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]

CROSSREFS

Cf. A054855.

Column k=4 of A245013. First differences of A046672.

Sequence in context: A194589 A189918 A164560 * A188161 A005178 A065315

Adjacent sequences:  A054851 A054852 A054853 * A054855 A054856 A054857

KEYWORD

easy,nonn

AUTHOR

Silvia Heubach (silvi(AT)cine.net), Apr 21 2000

STATUS

approved

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Last modified August 21 05:59 EDT 2017. Contains 290862 sequences.