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A322494 Number A(n,k) of tilings of a k X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation; square array A(n,k), n>=0, k>=0, read by antidiagonals. 11
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 8, 5, 1, 1, 1, 1, 8, 18, 18, 8, 1, 1, 1, 1, 13, 44, 68, 44, 13, 1, 1, 1, 1, 21, 107, 233, 233, 107, 21, 1, 1, 1, 1, 34, 257, 838, 1262, 838, 257, 34, 1, 1, 1, 1, 55, 621, 2989, 6523, 6523, 2989, 621, 55, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,13
COMMENTS
The shapes of the tiles are:
._.
._. | |
._. | | | |
._. | |_. | |_._. | |_._._.
|_| |___| |_____| |_______| ... .
.
The sequence of column k (or row k) satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.
LINKS
Wikipedia, Polyomino
EXAMPLE
A(3,3) = 8:
._____. ._____. ._____. ._____. ._____. ._____. ._____. ._____.
|_|_|_| | |_|_| |_|_|_| |_| |_| |_|_|_| |_| |_| | |_|_| | | |_|
|_|_|_| |___|_| | |_|_| |_|___| |_| |_| | |___| | |_|_| | |___|
|_|_|_| |_|_|_| |___|_| |_|_|_| |_|___| |___|_| |_____| |_____|.
.
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, 8, 18, 44, 107, 257, 621, ...
1, 1, 5, 18, 68, 233, 838, 2989, 10687, ...
1, 1, 8, 44, 233, 1262, 6523, 34468, 181615, ...
1, 1, 13, 107, 838, 6523, 51420, 396500, 3086898, ...
1, 1, 21, 257, 2989, 34468, 396500, 4577274, 52338705, ...
1, 1, 34, 621, 10687, 181615, 3086898, 52338705, 888837716, ...
MAPLE
b:= proc(n, l) option remember; local k, m, r;
if n=0 or l=[] then 1
elif min(l)>0 then (t-> b(n-t, map(h->h-t, l)))(min(l))
elif l[-1]=n then b(n, subsop(-1=[][], l))
else for k while l[k]>0 do od; r:= 0;
for m from 0 while k+m<=nops(l) and l[k+m]=0 and n>m do
r:= r+b(n, [l[1..k-1][], 1$m, m+1, l[k+m+1..nops(l)][]])
od; r
fi
end:
A:= (n, k)-> b(max(n, k), [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, l_] := b[n, l] = Module[{k, m, r}, Which[n == 0 || l == {}, 1, Min[l] > 0, Function[t, b[n-t, l-t]][Min[l]], l[[-1]] == n, b[n, ReplacePart[ l, -1 -> Nothing]], True, For[k=1, l[[k]] > 0, k++]; r = 0; For[m=0, k+m <= Length[l] && l[[k+m]] == 0 && n>m, m++, r = r + b[n, Join[l[[1 ;; k-1]], Array[1&, m], {m+1}, l[[k+m+1 ;; Length[l]]]]]]; r]];
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 2018, after Alois P. Heinz *)
CROSSREFS
Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A322496, A322497, A322498, A322499, A322500, A322501, A322502, A322503.
Main diagonal gives A322495.
Cf. A226444.
Sequence in context: A219924 A226444 A196929 * A258445 A129179 A120621
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 12 2018
STATUS
approved

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Last modified March 1 18:30 EST 2024. Contains 370443 sequences. (Running on oeis4.)