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%I #25 Nov 16 2018 12:28:30
%S 1,1,1,1,2,1,1,2,9,1,1,2,4,32,1,1,2,4,21,121,1,1,2,4,8,92,450,1,1,2,4,
%T 8,45,320,1681,1,1,2,4,8,16,248,1213,6272,1,1,2,4,8,16,93,1032,4822,
%U 23409,1,1,2,4,8,16,32,668,3524,18556,87362,1,1,2,4,8,16,32,189,3440,13173,70929,326041,1
%N Number A(n,k) of tilings of a k X k X n box using k*n bricks of shape k X 1 X 1; square array A(n,k), n>=0, k>=1, read by antidiagonals.
%H Alois P. Heinz, <a href="/A233308/b233308.txt">Antidiagonals n = 0..17, flattened</a>
%F A(n,k) = 2^n = A000079(n) for k>n.
%F A(n,n) = A068156(n) for n>1.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 2, 2, 2, ...
%e 1, 9, 4, 4, 4, 4, ...
%e 1, 32, 21, 8, 8, 8, ...
%e 1, 121, 92, 45, 16, 16, ...
%e 1, 450, 320, 248, 93, 32, ...
%e 1, 1681, 1213, 1032, 668, 189, ...
%e 1, 6272, 4822, 3524, 3440, 1832, ...
%e 1, 23409, 18556, 13173, 13728, 11976, ...
%p b:= proc(n, l) option remember; local d, t, k; d:= isqrt(nops(l));
%p if max(l[])>n then 0 elif n=0 then 1
%p elif min(l[])>0 then t:=min(l[]); b(n-t, map(x->x-t, l))
%p else for k while l[k]>0 do od; b(n, subsop(k=d, l))+
%p `if`(irem(k,d)=1 and {seq(l[k+j], j=1..d-1)}={0},
%p b(n, [seq(`if`(h-k<d and h-k>=0, 1, l[h]), h=1..nops(l))]), 0)+
%p `if`(k<=d and {seq(l[k+d*j], j=1..d-1)}={0},
%p b(n, [seq(`if`(irem(h-k, d)=0, 1, l[h]), h=1..nops(l))]), 0)
%p fi
%p end:
%p A:= (n, k)-> `if`(k>n, 2^n, b(n, [0$k^2])):
%p seq(seq(A(n, 1+d-n), n=0..d), d=0..11);
%t b[n_, l_] := b[n, l] = Module[{d, t, k}, d= Sqrt[Length[l]]; Which[ Max[l]>n, 0, n==0, 1, Min[l]>0, t=Min[l]; b[n-t, l-t], True, k=Position[l, 0, 1][[1, 1]]; b[n, ReplacePart[l, k->d]]+ If[Mod[k, d]==1 && Union[ Table[ l[[k+j]], {j, 1, d-1}]] == {0}, b[n, Table[ If [h-k<d && h-k>=0, 1, l[[h]] ], {h, 1, Length[l]}]], 0]+ If[k <= d && Union[ Table[ l[[k+d*j]], {j, 1, d-1}]] == {0}, b[n, Table[ If[ Mod[h-k, d] == 0, 1, l[[h]] ], {h, 1, Length[l]}]], 0] ] ]; a[n_, k_]:= If[k>n, 2^n, b[n, Array[0&, k^2]]]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 11}] // Flatten (* _Jean-François Alcover_, Dec 13 2013, translated from Maple *)
%Y Columns k=1-6 give: A000012, A006253, A233289, A233291, A233294, A233424.
%Y Diagonals include: A000079, A068156.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Dec 07 2013
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