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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. 16
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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.

Diagonal gives: A182106.

Sequence in context: A033630 A308608 A323719 * A101446 A333769 A259396

Adjacent sequences:  A220119 A220120 A220121 * A220123 A220124 A220125

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 05 2012

STATUS

approved

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Last modified May 14 19:53 EDT 2021. Contains 343903 sequences. (Running on oeis4.)