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A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals. 10
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 4, 3, 4, 2, 0, 1, 1, 0, 3, 8, 8, 8, 8, 3, 0, 1, 1, 1, 4, 13, 21, 28, 21, 13, 4, 1, 1, 1, 0, 5, 19, 31, 65, 65, 31, 19, 5, 0, 1, 1, 0, 7, 35, 70, 170, 267, 170, 70, 35, 7, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,25

LINKS

Alois P. Heinz, Antidiagonals n = 0..27, flattened

Wikipedia, Tromino

EXAMPLE

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

  ._._____.  ._____._.  ._._._._.

  | |_____|  |_____| |  | . | . |

  | | . | |  | | . | |  |___|___|

  |_|___| |  | |___|_|  | . | . |

  |_____|_|  |_|_____|  |___|___| .

Square array A(n,k) begins:

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

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

  1,  0,  1,  1,  1,   2,    2,     3,     4, ...

  1,  1,  1,  2,  3,   4,    8,    13,    19, ...

  1,  0,  1,  3,  3,   8,   21,    31,    70, ...

  1,  0,  2,  4,  8,  28,   65,   170,   456, ...

  1,  1,  2,  8, 21,  65,  267,   804,  2530, ...

  1,  0,  3, 13, 31, 170,  804,  2744, 12343, ...

  1,  0,  4, 19, 70, 456, 2530, 12343, 66653, ...

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))+

         `if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=2, k+1=2, 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..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], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 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, 14}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)

CROSSREFS

Columns (or rows) k=0-10 give: A000012, A079978, A000931(n+3), A219968, A202536, A219969, A219970, A219971, A219972, A219973, A219974.

Main diagonal gives: A219975.

Sequence in context: A006842 A299038 A273693 * A060505 A336727 A316101

Adjacent sequences:  A219964 A219965 A219966 * A219968 A219969 A219970

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 02 2012

STATUS

approved

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Last modified June 27 16:27 EDT 2022. Contains 354896 sequences. (Running on oeis4.)