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A226444
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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.
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9
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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
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OFFSET
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0,13
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COMMENTS
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An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.
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LINKS
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FORMULA
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The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013
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EXAMPLE
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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, ...
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MAPLE
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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<nops(l) and l[k+1]=0, b(n, subsop(k=1, k+1=2, l)), 0)
fi
end:
A:= (n, 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]
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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