login
The OEIS is supported by the many generous donors 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
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-François 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 A336727 A316101
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 | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 23 13:51 EDT 2024. Contains 371914 sequences. (Running on oeis4.)