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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A219967 Number A(n,k) of tilings of a k X n rectangle using straight (3 X 1) trominoes and 2 X 2 tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals. 10
1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 0, 2, 4, 3, 4, 2, 0, 1, 1, 0, 3, 8, 8, 8, 8, 3, 0, 1, 1, 1, 4, 13, 21, 28, 21, 13, 4, 1, 1, 1, 0, 5, 19, 31, 65, 65, 31, 19, 5, 0, 1, 1, 0, 7, 35, 70, 170, 267, 170, 70, 35, 7, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,25

LINKS

Alois P. Heinz, Antidiagonals n = 0..27, flattened

Wikipedia, Tromino

EXAMPLE

A(4,4) = 3, because there are 3 tilings of a 4 X 4 rectangle using straight (3 X 1) trominoes and 2 X 2 tiles:

._._____.  ._____._.  ._._._._.

| |_____|  |_____| |  | . | . |

| | . | |  | | . | |  |___|___|

|_|___| |  | |___|_|  | . | . |

|_____|_|  |_|_____|  |___|___|

Square array A(n,k) begins:

1,  1,  1,  1,  1,   1,    1,     1,     1, ...

1,  0,  0,  1,  0,   0,    1,     0,     0, ...

1,  0,  1,  1,  1,   2,    2,     3,     4, ...

1,  1,  1,  2,  3,   4,    8,    13,    19, ...

1,  0,  1,  3,  3,   8,   21,    31,    70, ...

1,  0,  2,  4,  8,  28,   65,   170,   456, ...

1,  1,  2,  8, 21,  65,  267,   804,  2530, ...

1,  0,  3, 13, 31, 170,  804,  2744, 12343, ...

1,  0,  4, 19, 70, 456, 2530, 12343, 66653, ...

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=3, l))+

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

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

            b(n, subsop(k=1, k+1=1, k+2=1, l)), 0)

      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_] := b[n, l] = Module[{ k, t}, If [Max[l] > n, 0, If[n == 0 || l == {}, 1, If[ Min[l] > 0 , t = Min[l]; b[n-t, l-t], k = Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k -> 3]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 2, k+1 -> 2}]], 0] + If[k+1 < Length[l] && l[[k+1]] == 0 && l[[k+2]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1, k+2 -> 1}]], 0] ] ] ] ]; 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-Fran├žois Alcover, Dec 16 2013, translated from Maple *)

CROSSREFS

Columns (or rows) k=0-10 give: A000012, A079978, A000931(n+3), A219968, A202536, A219969, A219970, A219971, A219972, A219973, A219974.

Main diagonal gives: A219975.

Sequence in context: A006842 A299038 A273693 * A060505 A316101 A211452

Adjacent sequences:  A219964 A219965 A219966 * A219968 A219969 A219970

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Dec 02 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 11 16:01 EST 2019. Contains 329019 sequences. (Running on oeis4.)