A226206 Number A(n,k) of tilings of a k X n rectangle using integer-sided square tiles of area > 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 3, 1, 3, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 1, 0, 1, 1, 5, 0, 7, 0, 5, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 7, 7, 0, 0, 0, 0, 0, 1

Offset

0,41

Links

Alois P. Heinz, Antidiagonals n = 0..34, flattened

Example

A(6,4) = A(4,6) = 3:
  ._._._._._._.   ._._._._._._.   ._._._._._._.
  |   |   |   |   |       |   |   |   |       |
  |___|___|___|   |       |___|   |___|       |
  |   |   |   |   |       |   |   |   |       |
  |___|___|___|   |_______|___|   |___|_______|  .
Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1,  1, 1,   1,  1,   1, ...
  1, 0, 0, 0, 0, 0,  0, 0,   0,  0,   0, ...
  1, 0, 1, 0, 1, 0,  1, 0,   1,  0,   1, ...
  1, 0, 0, 1, 0, 0,  1, 0,   0,  1,   0, ...
  1, 0, 1, 0, 2, 0,  3, 0,   5,  0,   8, ...
  1, 0, 0, 0, 0, 1,  2, 0,   0,  0,   1, ...
  1, 0, 1, 1, 3, 2,  7, 7,  16, 19,  40, ...
  1, 0, 0, 0, 0, 0,  7, 1,   0,  0,   2, ...
  1, 0, 1, 0, 5, 0, 16, 0,  48,  0, 160, ...
  1, 0, 0, 1, 0, 0, 19, 0,   0, 50,  17, ...
  1, 0, 1, 0, 8, 1, 40, 2, 160, 17, 796, ...
  ...

Maple

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then 0 elif n=0 or l=[] then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=0;
         for i from k+1 to nops(l) while l[i]=0 do s:=s+
           b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)])
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n])):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which [Max[l] > n, 0, n == 0 || l == {}, 1, Min[l] > 0, t = Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s = 0; For[i = k+1, i <= Length[l] && l[[i]] == 0, i++, s = s + b[n, Join [l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]] ]]]; s]]; a [n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

Crossrefs

Columns (or rows) k=0-12 give: A000012, A000007, A059841, A079978, A079977, A226369, A226370, A226371, A226372, A226373, A226374, A226375, A226376.

Main diagonal gives A347800.

Cf. A219924.

Sequence in context: A364042 A033764 A033784 * A350682 A375150 A319660

Adjacent sequences: A226203 A226204 A226205 * A226207 A226208 A226209

Keyword

nonn,tabl

Author

Alois P. Heinz, May 31 2013

Status

approved