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 A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals. 9
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 6, 5, 1, 1, 1, 1, 8, 13, 13, 8, 1, 1, 1, 1, 13, 28, 42, 28, 13, 1, 1, 1, 1, 21, 60, 126, 126, 60, 21, 1, 1, 1, 1, 34, 129, 387, 524, 387, 129, 34, 1, 1, 1, 1, 55, 277, 1180, 2229, 2229, 1180, 277, 55, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed. LINKS Alois P. Heinz, Antidiagonals n = 0..44, flattened FORMULA The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013 EXAMPLE A(3,3) = 6:   ._____.  ._____.  ._____.  ._____.  ._____.  ._____.   |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|   |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|   |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|. Square array A(n,k) begins:   1, 1,  1,   1,    1,     1,      1,       1,        1, ...   1, 1,  1,   1,    1,     1,      1,       1,        1, ...   1, 1,  2,   3,    5,     8,     13,      21,       34, ...   1, 1,  3,   6,   13,    28,     60,     129,      277, ...   1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...   1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...   1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...   1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...   1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ... 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=1, l))+         `if`(k b(max(n, k), [0\$min(n, k)]): seq(seq(A(n, d-n), n=0..d), d=0..14); [Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - N. J. A. Sloane, Aug 22 2013] MATHEMATICA b[n_, l_] := b[n, l] = Module[{k, t}, Which[Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 1]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]], 0] ] ]; a[n_, k_] := b[Max[n, k], Array[0&, Min[n, k]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *) CROSSREFS Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A002478, A105262, A219737(n-1) for n>2, A219738 (n-1) for n>2, A219739(n-1) for n>1, A219740(n-1) for n>2, A226543, A226544. Main diagonal gives A066864(n-1). See A219741 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013 Cf. A322494. Sequence in context: A189006 A245013 A219924 * A196929 A322494 A258445 Adjacent sequences:  A226441 A226442 A226443 * A226445 A226446 A226447 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jun 06 2013 STATUS approved

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Last modified May 9 13:09 EDT 2021. Contains 343742 sequences. (Running on oeis4.)