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 A219987 Number A(n,k) of tilings of a k X n rectangle using dominoes and right trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 5, 5, 1, 1, 1, 0, 11, 8, 11, 0, 1, 1, 1, 24, 55, 55, 24, 1, 1, 1, 0, 53, 140, 380, 140, 53, 0, 1, 1, 1, 117, 633, 2319, 2319, 633, 117, 1, 1, 1, 0, 258, 1984, 15171, 21272, 15171, 1984, 258, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 LINKS Alois P. Heinz, Antidiagonals n = 0..31, flattened Wikipedia, Domino Wikipedia, Tromino EXAMPLE A(3,3) = 8, because there are 8 tilings of a 3 X 3 rectangle using dominoes and right trominoes: .___._. .___._. .___._. .___._. |___| | |___| | |___| | |_. | | | ._|_| | | |_| | |___| | |_|_| |_|___| |_|___| |_|___| |_|___| ._.___. ._.___. ._.___. ._.___. | |___| | | ._| | |___| | |___| |___| | |_|_| | |_|_. | |_| | | |___|_| |___|_| |___|_| |___|_| Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 0, 1, 0, 1, 0, 1, 0, ... 1, 1, 2, 5, 11, 24, 53, 117, ... 1, 0, 5, 8, 55, 140, 633, 1984, ... 1, 1, 11, 55, 380, 2319, 15171, 96139, ... 1, 0, 24, 140, 2319, 21272, 262191, 2746048, ... 1, 1, 53, 633, 15171, 262191, 5350806, 100578811, ... 1, 0, 117, 1984, 96139, 2746048, 100578811, 3238675344, ... 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=2, l))+ `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, True, k++, If[l[[k]] == 0, Break[]]]; b[n, ReplacePart[l, k -> 2]] + 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 -> 1, k+1 -> 1}]] + 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}]] + b[n, ReplacePart[l, {k -> 2, k+1 -> 2, 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 05 2013, translated from Alois P. Heinz's Maple program *) PROG (Sage) from sage.combinat.tiling import TilingSolver, Polyomino def A(n, k): p = Polyomino([(0, 0), (0, 1)]) q = Polyomino([(0, 0), (0, 1), (1, 0)]) T = TilingSolver([p, q], box=[n, k], reusable=True, reflection=True) return T.number_of_solutions() # Ralf Stephan, May 21 2014 CROSSREFS Columns (or rows) k=0-10 give: A000012, A059841, A052980, A165716, A165791, A219988, A219989, A219990, A219991, A219992, A219993. Main diagonal gives: A219994. Sequence in context: A262124 A199954 A333580 * A077614 A336396 A280379 Adjacent sequences: A219984 A219985 A219986 * A219988 A219989 A219990 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Dec 02 2012 STATUS approved

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Last modified December 9 23:05 EST 2022. Contains 358710 sequences. (Running on oeis4.)