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A219866 Number A(n,k) of tilings of a k X n rectangle using dominoes and straight (3 X 1) trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals. 17
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 2, 7, 14, 7, 2, 1, 1, 2, 15, 41, 41, 15, 2, 1, 1, 3, 30, 143, 184, 143, 30, 3, 1, 1, 4, 60, 472, 1069, 1069, 472, 60, 4, 1, 1, 5, 123, 1562, 5624, 9612, 5624, 1562, 123, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

LINKS

Alois P. Heinz, Antidiagonals n = 0..22, flattened

EXAMPLE

A(2,3) = A(3,2) = 4, because there are 4 tilings of a 3 X 2 rectangle using dominoes and straight (3 X 1) trominoes:

  .___.   .___.   .___.   .___.

  | | |   |___|   | | |   |___|

  | | |   |___|   |_|_|   | | |

  |_|_|   |___|   |___|   |_|_|

Square array A(n,k) begins:

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

  1,  0,  1,    1,     1,      2,        2,         3, ...

  1,  1,  2,    4,     7,     15,       30,        60, ...

  1,  1,  4,   14,    41,    143,      472,      1562, ...

  1,  1,  7,   41,   184,   1069,     5624,     29907, ...

  1,  2, 15,  143,  1069,   9612,    82634,    707903, ...

  1,  2, 30,  472,  5624,  82634,  1143834,  15859323, ...

  1,  3, 60, 1562, 29907, 707903, 15859323, 354859954, ...

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;

         b(n, subsop(k=3, l))+ b(n, subsop(k=2, l))+

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

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

             b(n, subsop(k=1, k+1=1, k+2=1, 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..10);

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], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + b[n, ReplacePart[l, k -> 2]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0]]]]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-Fran├žois Alcover, Dec 16 2013, translated from Maple *)

CROSSREFS

Columns (or rows) k=0-10 give: A000012, A000931(n+3), A129682, A219867, A219862, A219868, A219869, A219870, A219871, A219872, A219873.

Main diagonal gives: A219874.

Sequence in context: A247342 A174547 A119326 * A333418 A212363 A212382

Adjacent sequences:  A219863 A219864 A219865 * A219867 A219868 A219869

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Nov 30 2012

STATUS

approved

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Last modified September 21 00:14 EDT 2021. Contains 347596 sequences. (Running on oeis4.)