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A219946 Number A(n,k) of tilings of a k X n rectangle using right trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 4, 4, 4, 4, 0, 1, 1, 0, 5, 0, 6, 0, 5, 0, 1, 1, 0, 6, 8, 16, 16, 8, 6, 0, 1, 1, 0, 13, 0, 37, 0, 37, 0, 13, 0, 1, 1, 0, 16, 16, 92, 136, 136, 92, 16, 16, 0, 1, 1, 0, 25, 0, 245, 0, 545, 0, 245, 0, 25, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,18

LINKS

Alois P. Heinz, Antidiagonals n = 0..35, flattened

Wikipedia, Tromino

EXAMPLE

A(4,4) = 6, because there are 6 tilings of a 4 X 4 rectangle using right trominoes and 2 X 2 tiles:

  .___.___. .___.___. .___.___. .___.___. .___.___. .___.___.

  | . | . | | ._|_. | | ._| . | | ._|_. | | ._|_. | | . |_. |

  |___|___| |_| . |_| |_| |___| |_| ._|_| |_|_. |_| |___| |_|

  | . | . | | |___| | | |___| | | |_| . | | . |_| | | |___| |

  |___|___| |___|___| |___|___| |___|___| |___|___| |___|___|

Square array A(n,k) begins:

  1,  1,  1,  1,   1,    1,     1,      1,       1,        1, ...

  1,  0,  0,  0,   0,    0,     0,      0,       0,        0, ...

  1,  0,  1,  2,   1,    4,     5,      6,      13,       16, ...

  1,  0,  2,  0,   4,    0,     8,      0,      16,        0, ...

  1,  0,  1,  4,   6,   16,    37,     92,     245,      560, ...

  1,  0,  4,  0,  16,    0,   136,      0,    1128,      384, ...

  1,  0,  5,  8,  37,  136,   545,   2376,   10534,    46824, ...

  1,  0,  6,  0,  92,    0,  2376,   5504,   71248,   253952, ...

  1,  0, 13, 16, 245, 1128, 10534,  71248,  652036,  5141408, ...

  1,  0, 16,  0, 560,  384, 46824, 253952, 5141408, 44013568, ...

MAPLE

b:= proc(n, l) option remember; local k, t;

      if max(l[])>n then 0 elif n=0 or l=[] then 1

    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))

    else for k do if l[k]=0 then break fi od;

         `if`(k>1 and l[k-1]=1, b(n, subsop(k=2, k-1=2, l)), 0)+

         `if`(k<nops(l) and l[k+1]=1, b(n, subsop(k=2, k+1=2, l)), 0)+

         `if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=2, k+1=2, l))+

            b(n, subsop(k=1, k+1=2, l))+b(n, subsop(k=2, k+1=1, l)), 0)+

         `if`(k+1<nops(l) and l[k+1]=0 and l[k+2]=0,

            b(n, subsop(k=2, k+1=2, k+2=2, l)), 0)

      fi

    end:

A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

b[n_, l_] := b[n, l] = Module[{k, t}, If[Max[l] > n , 0 , If [n == 0 || l == {}, 1 , If[Min[l] > 0, t = Min[l]; b[n-t, l-t], For[k = 1, k <= Length[l], k++, If[l[[k]] == 0 , Break[]]]; If[k > 1 && l[[k-1]] == 1, b[n, ReplacePart[l, {k -> 2, k-1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 1, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 1}]], 0]+If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2, k+2 -> 2}]], 0]]]]]; a[n_, _] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Nov 26 2013, translated from Alois P. Heinz's Maple program *)

CROSSREFS

Columns (or rows) k=0-10 give: A000012, A000007, A052947, A077957, A165799, A190759, A219947, A219948, A219949, A219950, A219951.

Main diagonal gives: A219952.

Sequence in context: A295181 A215573 A163537 * A117449 A004594 A124210

Adjacent sequences:  A219943 A219944 A219945 * A219947 A219948 A219949

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 01 2012

STATUS

approved

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Last modified April 25 21:06 EDT 2019. Contains 322461 sequences. (Running on oeis4.)