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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
OFFSET
0,13
COMMENTS
A(n,n) = A034856(n+2) for n>=2.
LINKS
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Nov 29 2014
STATUS
approved