login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A069283 A285337 A033630 * A101446 A259396 A219093

Adjacent sequences:  A220119 A220120 A220121 * A220123 A220124 A220125

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 05 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 19 08:25 EDT 2018. Contains 305581 sequences. (Running on oeis4.)