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).

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

1,1

Links

Table of n, a(n) for n=1..25.

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).

Formula

a(n) = (2*n - 12)*(-1)^n + (2/3)*((9-5*sqrt(3))*(1+sqrt(3))^n + (9+5*sqrt(3))*(1-sqrt(3))^n) + (n/sqrt(3))*((sqrt(3)-1)*(1+sqrt(3))^n+ (sqrt(3)+1)*(1-sqrt(3))^n).

G.f.: 2*x*(1+2*x)/((1+x)^2*(1-2*x-2*x^2)^2). - Colin Barker, Apr 30 2012

Example

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.

Mathematica

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}]

Crossrefs

Cf. A127864, A127866, A127867, A127868, A127869, A127870, A127871, A127872.

Sequence in context: A230701 A365693 A168605 * A199923 A230269 A278023

Adjacent sequences: A127862 A127863 A127864 * A127866 A127867 A127868

Keyword

easy,nonn

Author

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

Status

approved