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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. 19
 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 `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-François 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. Cf. A219987, A364457. 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 February 24 07:07 EST 2024. Contains 370294 sequences. (Running on oeis4.)