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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A226444 Number A(n,k) of tilings of a k X n rectangle using 1 X 1 squares and L-tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals. 8
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 5, 6, 5, 1, 1, 1, 1, 8, 13, 13, 8, 1, 1, 1, 1, 13, 28, 42, 28, 13, 1, 1, 1, 1, 21, 60, 126, 126, 60, 21, 1, 1, 1, 1, 34, 129, 387, 524, 387, 129, 34, 1, 1, 1, 1, 55, 277, 1180, 2229, 2229, 1180, 277, 55, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

LINKS

Alois P. Heinz, Antidiagonals n = 0..44, flattened

FORMULA

The k-th column satisfies a recurrence of order Fibonacci(k+1) [Zeilberger] - see links in A228285. - N. J. A. Sloane, Aug 22 2013

EXAMPLE

A(3,3) = 6:

._____.  ._____.  ._____.  ._____.  ._____.  ._____.

|_|_|_|  | |_|_|  |_|_|_|  |_| |_|  |_|_|_|  |_| |_|

|_|_|_|  |___|_|  | |_|_|  |_|___|  |_| |_|  | |___|

|_|_|_|  |_|_|_|  |___|_|  |_|_|_|  |_|___|  |___|_|.

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,   6,   13,    28,     60,     129,      277, ...

1, 1,  5,  13,   42,   126,    387,    1180,     3606, ...

1, 1,  8,  28,  126,   524,   2229,    9425,    39905, ...

1, 1, 13,  60,  387,  2229,  13322,   78661,   466288, ...

1, 1, 21, 129, 1180,  9425,  78661,  647252,  5350080, ...

1, 1, 34, 277, 3606, 39905, 466288, 5350080, 61758332, ...

MAPLE

b:= proc(n, l) option remember; local k, 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; b(n, subsop(k=1, l))+

        `if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=1, k+1=2, l)), 0)

      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);

[Zeilberger gives Maple code to find generating functions for the columns - see links in A228285. - N. J. A. Sloane, Aug 22 2013]

MATHEMATICA

b[n_, l_] := b[n, l] = Module[{k, 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]]; b[n, ReplacePart[l, k -> 1]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 2}]], 0] ] ]; 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-Fran├žois Alcover, Dec 18 2013, translated from Maple *)

CROSSREFS

Columns (or rows) k=0+1,2-10 give: A000012, A000045(n+1), A002478, A105262, A219737(n-1) for n>2, A219738 (n-1) for n>2, A219739(n-1) for n>1, A219740(n-1) for n>2, A226543, A226544.

Main diagonal gives A066864(n-1).

See A219741 for an array with very similar entries. - N. J. A. Sloane, Aug 22 2013

Sequence in context: A189006 A245013 A219924 * A196929 A258445 A129179

Adjacent sequences:  A226441 A226442 A226443 * A226445 A226446 A226447

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jun 06 2013

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 May 22 13:18 EDT 2017. Contains 286872 sequences.