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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, 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 (list; graph; refs; listen; history; text; internal format)
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 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: 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

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Last modified July 29 11:34 EDT 2021. Contains 346344 sequences. (Running on oeis4.)