|
| |
|
|
A251072
|
|
Number A(n,k) of tilings of a 3k X n rectangle using 3n k-ominoes of shape I; square array A(n,k), n>=0, k>=0, read by antidiagonals.
|
|
11
|
|
|
|
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 41, 1, 1, 1, 1, 1, 19, 281, 1, 1, 1, 1, 1, 1, 57, 1183, 1, 1, 1, 1, 1, 1, 26, 121, 6728, 1, 1, 1, 1, 1, 1, 1, 75, 783, 31529, 1, 1, 1, 1, 1, 1, 1, 34, 154, 2861, 167089, 1, 1, 1, 1, 1, 1, 1, 1, 95, 269, 8133, 817991, 1, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,13
|
|
|
COMMENTS
|
A(n,n) = A034856(n+2) for n>=2.
|
|
|
LINKS
|
Alois P. Heinz, Antidiagonals n = 0..35, flattened
Wikipedia, Polyomino
|
|
|
EXAMPLE
|
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, 13, 1, 1, 1, 1, 1, 1, ...
1, 1, 41, 19, 1, 1, 1, 1, 1, ...
1, 1, 281, 57, 26, 1, 1, 1, 1, ...
1, 1, 1183, 121, 75, 34, 1, 1, 1, ...
1, 1, 6728, 783, 154, 95, 43, 1, 1, ...
1, 1, 31529, 2861, 269, 190, 117, 53, 1, ...
1, 1, 167089, 8133, 1732, 325, 229, 141, 64, ...
|
|
|
MAPLE
|
b:= proc(n, l) option remember; local d, k; d:= nops(l)/3;
if n=0 then 1
elif min(l[])>0 then (m->b(n-m, map(x->x-m, l)))(min(l[]))
else for k while l[k]>0 do od;
`if`(n<d, 0, b(n, subsop(k=d, l)))+
`if`(d=1 or k>2*d+1 or max(l[k..k+d-1][])>0, 0,
b(n, [l[1..k-1][], 1$d, l[k+d..3*d][]]))
fi
end:
A:= (n, k)-> `if`(k=0, 1, b(n, [0$3*k])):
seq(seq(A(n, d-n), n=0..d), d=0..12);
|
|
|
MATHEMATICA
|
b[n_, l_List] := b[n, l] = Module[{d = Length[l]/3, k}, Which[n == 0, 1, Min[l] > 0, Function[{m}, b[n-m, l-m]][Min[l]], True, For[k=1, l[[k]] > 0 , k++]; If[n<d, 0, b[n, ReplacePart[l, k -> d]]] + If[d == 1 || k > 2d + 1 || Max[l[[k ;; k + d - 1]]] > 0, 0, b[n, Join[l[[1 ;; k-1]], Array[1&, d], l[[k+d ;; 3*d]]]]]]]; A[n_, k_] := If[k == 0, 1, b[n, Array[0&, 3k]]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 30 2015, after Alois P. Heinz *)
|
|
|
CROSSREFS
|
Columns k=0+1,2-10 give: A000012, A028468, A251073, A251074, A247218, A251075, A251076, A251077, A251078, A251079.
Cf. A034856, A250662.
Sequence in context: A165400 A181154 A357912 * A353805 A332018 A010227
Adjacent sequences: A251069 A251070 A251071 * A251073 A251074 A251075
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
Alois P. Heinz, Nov 29 2014
|
|
|
STATUS
|
approved
|
| |
|
|
