A322494 Number A(n,k) of tilings of a k X n rectangle using V (2m+1)-ominoes (m >= 0) in standard orientation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 8, 5, 1, 1, 1, 1, 8, 18, 18, 8, 1, 1, 1, 1, 13, 44, 68, 44, 13, 1, 1, 1, 1, 21, 107, 233, 233, 107, 21, 1, 1, 1, 1, 34, 257, 838, 1262, 838, 257, 34, 1, 1, 1, 1, 55, 621, 2989, 6523, 6523, 2989, 621, 55, 1, 1

Offset

0,13

Comments

The shapes of the tiles are:

._.

._. | |

._. | | | |

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

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

.

The sequence of column k (or row k) satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

Links

Alois P. Heinz, Antidiagonals n = 0..23, flattened

Wikipedia, Polyomino

Example

A(3,3) = 8:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|  | |_|_|  | | |_|
  |_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|  | |_|_|  | |___|
  |_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|  |_____|  |_____|.
.
Square array A(n,k) begins:
  1, 1,  1,   1,     1,      1,       1,        1,         1, ...
  1, 1,  1,   1,     1,      1,       1,        1,         1, ...
  1, 1,  2,   3,     5,      8,      13,       21,        34, ...
  1, 1,  3,   8,    18,     44,     107,      257,       621, ...
  1, 1,  5,  18,    68,    233,     838,     2989,     10687, ...
  1, 1,  8,  44,   233,   1262,    6523,    34468,    181615, ...
  1, 1, 13, 107,   838,   6523,   51420,   396500,   3086898, ...
  1, 1, 21, 257,  2989,  34468,  396500,  4577274,  52338705, ...
  1, 1, 34, 621, 10687, 181615, 3086898, 52338705, 888837716, ...

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$m, m+1, l[k+m+1..nops(l)][]])
         od; r
      fi
    end:
A:= (n, k)-> b(max(n, k), [0$min(n, k)]):
seq(seq(A(n, d-n), n=0..d), d=0..14);

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_, k_] := b[Max[n, k], Array[0&, Min[n, k]]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 18 2018, after Alois P. Heinz *)

Crossrefs

Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A322496, A322497, A322498, A322499, A322500, A322501, A322502, A322503.

Main diagonal gives A322495.

Cf. A226444.

Sequence in context: A219924 A226444 A196929 * A258445 A129179 A120621

Adjacent sequences: A322491 A322492 A322493 * A322495 A322496 A322497

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 12 2018

Status

approved