A245965 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 1 X 1 tiles.
1, 0, 0, 1, 0, 4, 0, 0, 1, 2, 0, 0, 8, 0, 0, 1, 0, 0, 20, 0, 0, 12, 0, 0, 1, 0, 16, 0, 0, 54, 0, 0, 16, 0, 0, 1, 4, 0, 0, 112, 0, 0, 104, 0, 0, 20, 0, 0, 1, 0, 0, 108, 0, 0, 352, 0, 0, 170, 0, 0, 24, 0, 0, 1, 0, 48, 0, 0, 664, 0, 0, 800, 0, 0, 252, 0, 0, 28, 0, 0, 1, 8, 0, 0, 704, 0, 0, 2280, 0, 0, 1520, 0, 0, 350, 0, 0, 32, 0, 0, 1, 0, 0, 416, 0, 0, 4064, 0, 0, 5820, 0, 0, 2576, 0, 0, 464, 0, 0, 36, 0, 0, 1
Offset: 0
Examples
T(2,1)=4 because we can place the 1 X 1 tile in any corner of the 2 X 2 board. Triangle starts: 1; 0, 0, 1; 0, 4, 0, 0, 1; 2, 0, 0, 8, 0, 0, 1; 0, 0, 20, 0, 0, 12, 0, 0, 1;
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
Programs
-
Maple
G := 1/(1-t^2*z-4*t*z^2-2*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 .. 2*j) end do; # yields sequence in triangular form
Formula
G.f.: 1/(1-t^2*z - 4*t*z^2 - 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).
From Robert Israel, Aug 15 2014: (Start)
T(n+3,k+2) = T(n+2,k) + 4*T(n+1,k+1) + 2*T(n,k+2).
T(n,0) = 2^(n/3) if n == 0 (mod 3), T(n,0) = 0 otherwise.
T(n,1) = (n+1)*2^((n+4)/3)/3 if n == 2 (mod 3), T(n,1) = 0 otherwise.
(End)
Comments