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.

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

Offset: 0

Author

Alois P. Heinz, Mar 13 2014

Keywords

• nonn
• tabl

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.

Examples

			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, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..36, flattened

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.

Cf. A099390, A187616, A187617, A187596, A220644.

Programs

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`(k1 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 `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[k1 && 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 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 *)