

A245966


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


1



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
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OFFSET

0,4


COMMENTS

Row n contains 1+floor(2n/3) entries.
Sum of entries in row n = A127864(n).
Sum(k*T(n,k), k>=0) = 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 1x1 tiles, and s marking Lshaped 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 Lshaped 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/(1z4*t*z^22*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: A080102 A106475 A134829 * A130297 A271478 A112032
Adjacent sequences: A245963 A245964 A245965 * A245967 A245968 A245969


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Aug 15 2014


STATUS

approved



