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A239264 Number A(n,k) of domicule tilings of a k X n grid; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 3, 0, 1, 1, 1, 5, 5, 1, 1, 1, 0, 11, 0, 11, 0, 1, 1, 1, 21, 43, 43, 21, 1, 1, 1, 0, 43, 0, 280, 0, 43, 0, 1, 1, 1, 85, 451, 1563, 1563, 451, 85, 1, 1, 1, 0, 171, 0, 9415, 0, 9415, 0, 171, 0, 1, 1, 1, 341, 4945, 55553, 162409, 162409, 55553, 4945, 341, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,13
COMMENTS
A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.
LINKS
EXAMPLE
A(3,2) = 5:
+-----+ +-----+ +-----+ +-----+ +-----+
|o o-o| |o o o| |o o o| |o o o| |o-o o|
|| | || X | || | || | X || | ||
|o o-o| |o o o| |o o o| |o o o| |o-o o|
+-----+ +-----+ +-----+ +-----+ +-----+
A(4,3) = 43:
+-------+ +-------+ +-------+ +-------+ +-------+
|o o o o| |o o o-o| |o o-o o| |o o-o o| |o o-o o|
|| X || | X | | \ / | || || | \ ||
|o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
| | | X | || || | \ \ | || \ |
|o-o o-o| |o-o o o| |o o-o o| |o-o o o| |o o-o o|
+-------+ +-------+ +-------+ +-------+ +-------+ ...
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 0, 1, 0, 1, ...
1, 1, 3, 5, 11, 21, 43, ...
1, 0, 5, 0, 43, 0, 451, ...
1, 1, 11, 43, 280, 1563, 9415, ...
1, 0, 21, 0, 1563, 0, 162409, ...
1, 1, 43, 451, 9415, 162409, 3037561, ...
MAPLE
b:= proc(n, l) option remember; local d, f, k;
d:= nops(l)/2; f:=false;
if n=0 then 1
elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
else for k to d while not l[k] do od;
`if`(k<d and n>1 and l[k+d+1],
b(n, subsop(k=f, k+d+1=f, l)), 0)+
`if`(k>1 and n>1 and l[k+d-1],
b(n, subsop(k=f, k+d-1=f, l)), 0)+
`if`(n>1 and l[k+d], b(n, subsop(k=f, k+d=f, l)), 0)+
`if`(k<d and l[k+1], b(n, subsop(k=f, k+1=f, l)), 0)
fi
end:
A:= (n, k)-> `if`(irem(n*k, 2)>0, 0,
`if`(k>n, A(k, n), b(n, [true$(k*2)]))):
seq(seq(A(n, d-n), n=0..d), d=0..14);
MATHEMATICA
b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which [n == 0, 1, l[[1 ;; d]] == Array[f&, d], b[n-1, Join[l[[d+1 ;; 2*d]], Array[True&, d]]], True, For[k=1, !l[[k]], k++]; If[k<d && n>1 && l[[k+d+1]], b[n, ReplacePart[l, {k -> f, k+d+1 -> f}]], 0] + If[k>1 && n>1 && l[[k+d-1]], b[n, ReplacePart[l, {k -> f, k+d-1 -> f}]], 0] + If[n>1 && l[[k+d]], b[n, ReplacePart[l, {k -> f, k+d -> f}]], 0] + If[k<d && l[[k+1]], b[n, ReplacePart[l, {k -> f, k+1 -> f}]], 0]]]; A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Feb 02 2015, after Alois P. Heinz *)
CROSSREFS
Columns (or rows) k=0-10 give: A000012, A059841, A001045(n+1), A239265, A239266, A239267, A239268, A239269, A239270, A239271, A239272.
Bisection of main diagonal gives: A239273.
Sequence in context: A338940 A048838 A181872 * A294289 A059341 A249442
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 13 2014
STATUS
approved

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Last modified July 18 13:57 EDT 2024. Contains 374378 sequences. (Running on oeis4.)