A220122 Number A(n,k) of tilings of a k X n rectangle using integer-sided rectangular tiles of area k; 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, 1, 3, 1, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 1, 3, 3, 8, 1, 1, 1, 1, 2, 1, 9, 4, 13, 1, 1, 1, 1, 1, 4, 1, 16, 6, 21, 1, 1, 1, 1, 2, 1, 7, 2, 35, 9, 34, 1, 1, 1, 1, 1, 3, 1, 13, 3, 65, 13, 55, 1, 1, 1, 1, 2, 2, 9, 1, 46, 4, 143, 19, 89, 1, 1

Offset

0,13

Comments

Row n gives: 1 followed by period A003418(n): (1, A000045(n+1), ...) repeated; offset 0.

Links

Alois P. Heinz, Antidiagonals n = 0..32, flattened

Formula

For prime p column p has g.f.: 1/(1-x-x^p) or a_p(n) = Sum_{j=0..floor(n/p)} C(n-(p-1)*j,j).

Example

A(4,4) = 9, because there are 9 tilings of a 4 X 4 rectangle using integer-sided rectangular tiles of area 4:
._._._._.  ._______.  .___.___.  ._.___._.  ._______.
| | | | |  |_______|  |   |   |  | |   | |  |_______|
| | | | |  |_______|  |___|___|  | |___| |  |   |   |
| | | | |  |_______|  |   |   |  | |   | |  |___|___|
|_|_|_|_|  |_______|  |___|___|  |_|___|_|  |_______|
._._.___.  ._______.  .___._._.  .___.___.
| | |   |  |_______|  |   | | |  |   |   |
| | |___|  |_______|  |___| | |  |___|___|
| | |   |  |   |   |  |   | | |  |_______|
|_|_|___|  |___|___|  |___|_|_|  |_______|
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,  1,   1, ...
1, 1,  2,  1,   2, 1,    2, 1,    2,  1,   2, ...
1, 1,  3,  2,   3, 1,    4, 1,    3,  2,   3, ...
1, 1,  5,  3,   9, 1,    7, 1,    9,  3,   5, ...
1, 1,  8,  4,  16, 2,   13, 1,   16,  4,   9, ...
1, 1, 13,  6,  35, 3,   46, 1,   35,  6,  15, ...
1, 1, 21,  9,  65, 4,   88, 2,   65,  9,  26, ...
1, 1, 34, 13, 143, 5,  209, 3,  250, 13,  44, ...
1, 1, 55, 19, 281, 6,  473, 4,  495, 37,  75, ...
1, 1, 89, 28, 590, 8, 1002, 5, 1209, 64, 254, ...

Maple

b:= proc(n, l) option remember; local i, k, m, 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, m:=0, nops(l);
         for i from k to m while l[i]=0 do if irem(m, 1+i-k, 'q')=0
           and q<=n then s:= s+ b(n, [l[j]$j=1..k-1, q$j=k..i,
           l[j]$j=i+1..m]) fi od; s
      fi
    end:
A:= (n, k)-> b(n, [0$k]):
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

b[n_, l_] := b[n, l] = Module[{i, k, m, 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, m} = {0, Length[l]}; For[ i = k , i <= m && l[[i]] == 0, i++, If[Mod[m, 1+i-k ] == 0 && (q = Quotient[m, 1+i-k]) <= n, s = s+b[n, Join[ l[[1 ;; k-1]], Array[q &, i-k+1], l[[i+1 ;; m]] ]]]]; s]]; a[n_, k_] := b[n, Array[0&, k]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

Crossrefs

Columns k=0+1, 2-11, 13 give: A000012, A000045(n+1), A000930, A220123, A003520, A220124, A005709, A220125, A220126, A220127, A017905(n+11), A017907(n+13).

Rows n=0+1, 2-10 give: A000012, A040001, A220128, A220129, A220130, A220131, A220132, A220133, A220134, A220135.

Main diagonal gives: A182106.

Sequence in context: A371213 A378604 A323719 * A101446 A333769 A259396

Adjacent sequences: A220119 A220120 A220121 * A220123 A220124 A220125

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 05 2012

Status

approved