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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.
10
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
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
Liang Kai, Antidiagonals n = 0..75, flattened (antidiagonals 0..44 from Alois P. Heinz)
Kai Liang, Independent Set Enumeration and Estimation of Related Constants of Grid Graphs and Their Variants, arXiv:2507.04007 [math.CO], 2025. See p. 13.
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<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]
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.
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 06 2013
STATUS
approved