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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

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

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-Fran├žois 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.

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

Sequence in context: A338940 A048838 A181872 * A294289 A059341 A249442

Adjacent sequences:  A239261 A239262 A239263 * A239265 A239266 A239267

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 13 2014

STATUS

approved

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Last modified August 11 20:13 EDT 2022. Contains 356067 sequences. (Running on oeis4.)