OFFSET
0,13
LINKS
Alois P. Heinz, Antidiagonals n = 0..29, flattened
Wikipedia, Tromino
EXAMPLE
A(2,2) = 5, because there are 5 tilings of a 2 X 2 rectangle using right trominoes and 1 X 1 tiles:
._._. ._._. .___. .___. ._._.
|_|_| | |_| | ._| |_. | |_| |
|_|_| |___| |_|_| |_|_| |___|
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 5, 11, 33, 87, 241, 655, ...
1, 1, 11, 39, 195, 849, 3895, 17511, ...
1, 1, 33, 195, 2023, 16839, 151817, 1328849, ...
1, 1, 87, 849, 16839, 249651, 4134881, 65564239, ...
1, 1, 241, 3895, 151817, 4134881, 128938297, 3814023955, ...
1, 1, 655, 17511, 1328849, 65564239, 3814023955, 207866584389, ...
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=1, l))+
`if`(k>1 and l[k-1]=1, b(n, subsop(k=2, k-1=2, l)), 0)+
`if`(k<nops(l) and l[k+1]=1, b(n, subsop(k=2, k+1=2, l)), 0)+
`if`(k<nops(l) and l[k+1]=0, b(n, subsop(k=1, k+1=2, l))+
b(n, subsop(k=2, k+1=1, l))+ 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=2, k+1=2, k+2=2, 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}, Which[ Max[l] > n , 0, n == 0 || l == {} , 1 , Min[l] > 0 , t := Min[l]; b[n - t, l - t] , True, For[k = 1, True, k++, If[ l[[k]] == 0 , Break[] ] ]; b[n, ReplacePart[l, k -> 1]] + If[k > 1 && l[[k - 1]] == 1, b[n, ReplacePart[l, {k -> 2, k - 1 -> 2}]], 0] + If[k < Length[l] && l[[k + 1]] == 1, b[n, ReplacePart[l, {k -> 2, k + 1 -> 2}]], 0] + If[k < Length[l] && l[[k + 1]] == 0, b[n, ReplacePart[l, {k -> 1, k + 1 -> 2}]] + b[n, ReplacePart[l, {k -> 2, k + 1 -> 1}]] + 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 -> 2, k + 1 -> 2, k + 2 -> 2}]], 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 09 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Dec 03 2012
STATUS
approved