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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 `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-François 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 June 26 12:46 EDT 2022. Contains 354883 sequences. (Running on oeis4.)