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A219866 Number A(n,k) of tilings of a k X n rectangle using dominoes and straight (3 X 1) trominoes; square array A(n,k), n>=0, k>=0, read by antidiagonals. 19
1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 4, 4, 1, 1, 1, 2, 7, 14, 7, 2, 1, 1, 2, 15, 41, 41, 15, 2, 1, 1, 3, 30, 143, 184, 143, 30, 3, 1, 1, 4, 60, 472, 1069, 1069, 472, 60, 4, 1, 1, 5, 123, 1562, 5624, 9612, 5624, 1562, 123, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,13
LINKS
EXAMPLE
A(2,3) = A(3,2) = 4, because there are 4 tilings of a 3 X 2 rectangle using dominoes and straight (3 X 1) trominoes:
.___. .___. .___. .___.
| | | |___| | | | |___|
| | | |___| |_|_| | | |
|_|_| |___| |___| |_|_|
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 1, 1, 1, 2, 2, 3, ...
1, 1, 2, 4, 7, 15, 30, 60, ...
1, 1, 4, 14, 41, 143, 472, 1562, ...
1, 1, 7, 41, 184, 1069, 5624, 29907, ...
1, 2, 15, 143, 1069, 9612, 82634, 707903, ...
1, 2, 30, 472, 5624, 82634, 1143834, 15859323, ...
1, 3, 60, 1562, 29907, 707903, 15859323, 354859954, ...
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))+ b(n, subsop(k=2, l))+
`if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=1, k+1=1, 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..10);
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]] + b[n, ReplacePart[l, k -> 2]] + If[k < Length[l] && l[[k+1]] == 0, b[n, ReplacePart[l, {k -> 1, k+1 -> 1}]], 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, 10}] // Flatten (* Jean-François Alcover, Dec 16 2013, translated from Maple *)
CROSSREFS
Columns (or rows) k=0-10 give: A000012, A000931(n+3), A129682, A219867, A219862, A219868, A219869, A219870, A219871, A219872, A219873.
Main diagonal gives: A219874.
Sequence in context: A247342 A174547 A119326 * A333418 A212363 A212382
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 30 2012
STATUS
approved

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Last modified February 24 07:07 EST 2024. Contains 370294 sequences. (Running on oeis4.)