A054854 Number of ways to tile a 4 X n region with 1 X 1 and 2 X 2 tiles.

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

Offset

0,3

Links

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

D. Abadie, J. Andreoli, S. Egan, T. Reddy, D. Xiong, Y. Zhao, and A. Zhu, On the Number of Tilings of a 4-By-N Rectangle with 1-By-1 and 2-By-2 Squares, Girls' Angle Bulletin, Vol. 15, No. 2 (2021), 8-12.

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.

Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 7.

R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.1.

A. Ugolnikova, Pavages Aleatories, Phd Thesis (2016) Section 2.2.3

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). [See proofs in Mathar (a transfer matrix approach) and in Abadie et al.(direct proof).] - 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

a(n) = F(n+1)^2 + Sum_{k=1..n-1} F(k)^2 * a(n-k-1), for n >= 0, where F(k) = A000045(k) (Fibonacci numbers), see Abadie, et al. - Richard S. Chang, Jan 21 2022

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]
(* Second program: *)
LinearRecurrence[{2, 3, -2}, {1, 1, 5}, 30] (* Jean-François Alcover, Jul 28 2018 *)

Crossrefs

Cf. A054855.

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

Sequence in context: A189918 A318415 A164560 * A188161 A323352 A005178

Adjacent sequences: A054851 A054852 A054853 * A054855 A054856 A054857

Keyword

easy,nonn

Author

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

Status

approved