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

0,4

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).

Links

Table of n, a(n) for n=0..60.

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

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).

Example

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;

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

Crossrefs

Cf. A127864, A127865, A245965.

Sequence in context: A106475 A370614 A134829 * A130297 A271478 A112032

Adjacent sequences: A245963 A245964 A245965 * A245967 A245968 A245969

Keyword

nonn,tabf

Author

Emeric Deutsch, Aug 15 2014

Status

approved