A245966 Triangle read by rows: T(n,k) is the number of tilings of a 2 X n board with 1 X 1 and L-shaped tiles (where the L-shaped tiles cover 3 squares) that have k L-shaped tiles.

1, 1, 1, 4, 1, 8, 2, 1, 12, 20, 1, 16, 54, 16, 1, 20, 104, 112, 4, 1, 24, 170, 352, 108, 1, 28, 252, 800, 664, 48, 1, 32, 350, 1520, 2280, 704, 8, 1, 36, 464, 2576, 5820, 4064, 416, 1, 40, 594, 4032, 12404, 14784, 4560, 128, 1, 44, 740, 5952, 23408, 41104, 25376, 3200, 16

Offset: 0

Author

Emeric Deutsch, Aug 15 2014

Keywords

• nonn
• tabf

Comments

Row n contains 1+floor(2n/3) entries.

Sum of entries in row n = A127864(n).

Sum_{k>=0} k*T(n,k) = A127866(n).

Examples

			T(2,1) = 4 because we can place the L-shaped tile in the 2*2 board in 4 positions.
Triangle starts:
  1;
  1;
  1,  4;
  1,  8,  2;
  1, 12, 20;
  1, 16, 54, 16;
		

Links

• P. Chinn, R. Grimaldi and S. Heubach, Tiling with L's and Squares, Journal of Integer Sequences, Vol. 10 (2007), Article 07.2.8

Crossrefs

Cf. A127864, A127865, A245965.

Programs

Maple

G := 1/(1-z-4*t*z^2-2*t^2*z^3): Gser := simplify(series(G, z = 0, 15)): for j from 0 to 13 do P[j] := sort(coeff(Gser, z, j)) end do: for j from 0 to 13 do seq(coeff(P[j], t, i), i = 0 .. floor(2*j*(1/3))) end do; # yields sequence in triangular form

Formula

G.f.: 1/(1 - z - 4*t*z^2 - 2*t^2*z^3).

The trivariate g.f. with z marking length, t marking 1 X 1 tiles, and s marking L-shaped tiles is 1/(1 - t^2*z - 4*t*s*z^2 - 2*s^2*z^3).