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).

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

2,1

Links

Table of n, a(n) for n=2..26.

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).

Formula

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].

G.f.: 4x^2/((1+x)(1-2x-2x^2)^2).

Example

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.

Mathematica

Table[Coefficient[Normal[Series[4x^2/((1 + x)(1 - 2x - 2x^2)^2), {x, 0, 20}]], x, n], {n, 0, 20}]

Crossrefs

Cf. A127864, A127865, A127867, A127868, A127869, A127870, A127871, A127872.

Sequence in context: A025282 A303392 A149403 * A149404 A149405 A149406

Adjacent sequences: A127863 A127864 A127865 * A127867 A127868 A127869

Keyword

easy,nonn

Author

Silvia Heubach (sheubac(AT)calstatela.edu), Feb 03 2007

Extensions

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.

Status

approved