cp's OEIS Frontend

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.

%I A239264 #25 Nov 23 2018 06:53:00
%S A239264 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,
%T A239264 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,
%U A239264 171,0,9415,0,9415,0,171,0,1,1,1,341,4945,55553,162409,162409,55553,4945,341,1,1
%N 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.
%C A239264 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.
%H A239264 Alois P. Heinz, <a href="/A239264/b239264.txt">Antidiagonals n = 0..36, flattened</a>
%e A239264 A(3,2) = 5:
%e A239264   +-----+ +-----+ +-----+ +-----+ +-----+
%e A239264   |o o-o| |o o o| |o o o| |o o o| |o-o o|
%e A239264   ||    | ||  X | || | || | X  || |    ||
%e A239264   |o o-o| |o o o| |o o o| |o o o| |o-o o|
%e A239264   +-----+ +-----+ +-----+ +-----+ +-----+
%e A239264 A(4,3) = 43:
%e A239264   +-------+ +-------+ +-------+ +-------+ +-------+
%e A239264   |o o o o| |o o o-o| |o o-o o| |o o-o o| |o o-o o|
%e A239264   ||  X  || | X     | | \   / | ||     || | \    ||
%e A239264   |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
%e A239264   |       | |     X | ||     || |   \ \ | ||    \ |
%e A239264   |o-o o-o| |o-o o o| |o o-o o| |o-o o o| |o o-o o|
%e A239264   +-------+ +-------+ +-------+ +-------+ +-------+ ...
%e A239264 Square array A(n,k) begins:
%e A239264   1, 1,  1,   1,    1,      1,       1, ...
%e A239264   1, 0,  1,   0,    1,      0,       1, ...
%e A239264   1, 1,  3,   5,   11,     21,      43, ...
%e A239264   1, 0,  5,   0,   43,      0,     451, ...
%e A239264   1, 1, 11,  43,  280,   1563,    9415, ...
%e A239264   1, 0, 21,   0, 1563,      0,  162409, ...
%e A239264   1, 1, 43, 451, 9415, 162409, 3037561, ...
%p A239264 b:= proc(n, l) option remember; local d, f, k;
%p A239264       d:= nops(l)/2; f:=false;
%p A239264       if n=0 then 1
%p A239264     elif l[1..d]=[f$d] then b(n-1, [l[d+1..2*d][], true$d])
%p A239264     else for k to d while not l[k] do od;
%p A239264          `if`(k<d and n>1 and l[k+d+1],
%p A239264                               b(n, subsop(k=f, k+d+1=f, l)), 0)+
%p A239264          `if`(k>1 and n>1 and l[k+d-1],
%p A239264                               b(n, subsop(k=f, k+d-1=f, l)), 0)+
%p A239264          `if`(n>1 and l[k+d], b(n, subsop(k=f, k+d=f, l)), 0)+
%p A239264          `if`(k<d and l[k+1], b(n, subsop(k=f, k+1=f, l)), 0)
%p A239264       fi
%p A239264     end:
%p A239264 A:= (n, k)-> `if`(irem(n*k, 2)>0, 0,
%p A239264     `if`(k>n, A(k, n), b(n, [true$(k*2)]))):
%p A239264 seq(seq(A(n, d-n), n=0..d), d=0..14);
%t A239264 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_ *)
%Y A239264 Columns (or rows) k=0-10 give: A000012, A059841, A001045(n+1), A239265, A239266, A239267, A239268, A239269, A239270, A239271, A239272.
%Y A239264 Bisection of main diagonal gives: A239273.
%Y A239264 Cf. A099390, A187616, A187617, A187596, A220644.
%K A239264 nonn,tabl
%O A239264 0,13
%A A239264 _Alois P. Heinz_, Mar 13 2014