A322495 Number of tilings of an n X n square using V (2m+1)-ominoes (m >= 0) in standard orientation.

1, 1, 2, 8, 68, 1262, 51420, 4577274, 888837716, 376116199534, 346688563051200, 695975307003529228

Offset

0,3

Comments

The shapes of the tiles are:

._.

._. | |

._. | | | |

._. | |_. | |_._. | |_._._.

|_| |___| |_____| |_______| ... .

Links

Table of n, a(n) for n=0..11.

Wikipedia, Polyomino

Example

a(3) = 8:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|  | |_|_|  | | |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|  | |_|_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|  |_____|  |_____|.
.

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$m, m+1, l[k+m+1..nops(l)][]])
         od; r
      fi
    end:
a:= n-> b(n, [0$n]):
seq(a(n), n=0..9);

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_] := b[n, Array[0&, n]];
a /@ Range[0, 9] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)

Crossrefs

Main diagonal of A322494.

Sequence in context: A157752 A055547 A113087 * A332637 A239843 A099729

Adjacent sequences: A322492 A322493 A322494 * A322496 A322497 A322498

Keyword

nonn,more

Author

Alois P. Heinz, Dec 12 2018

Status

approved