A233308 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.

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, 8, 45, 320, 1681, 1, 1, 2, 4, 8, 16, 248, 1213, 6272, 1, 1, 2, 4, 8, 16, 93, 1032, 4822, 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

Offset

0,5

Links

Alois P. Heinz, Antidiagonals n = 0..17, flattened

Formula

A(n,k) = 2^n = A000079(n) for k>n.

A(n,n) = A068156(n) for n>1.

Example

Square array A(n,k) begins:
  1,     1,     1,     1,     1,     1, ...
  1,     2,     2,     2,     2,     2, ...
  1,     9,     4,     4,     4,     4, ...
  1,    32,    21,     8,     8,     8, ...
  1,   121,    92,    45,    16,    16, ...
  1,   450,   320,   248,    93,    32, ...
  1,  1681,  1213,  1032,   668,   189, ...
  1,  6272,  4822,  3524,  3440,  1832, ...
  1, 23409, 18556, 13173, 13728, 11976, ...

Maple

b:= proc(n, l) option remember; local d, t, k; d:= isqrt(nops(l));
      if max(l[])>n then 0 elif n=0 then 1
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(x->x-t, l))
    else for k while l[k]>0 do od; b(n, subsop(k=d, l))+
         `if`(irem(k, d)=1 and {seq(l[k+j], j=1..d-1)}={0},
         b(n, [seq(`if`(h-k<d and h-k>=0, 1, l[h]), h=1..nops(l))]), 0)+
         `if`(k<=d and {seq(l[k+d*j], j=1..d-1)}={0},
         b(n, [seq(`if`(irem(h-k, d)=0, 1, l[h]), h=1..nops(l))]), 0)
      fi
    end:
A:= (n, k)-> `if`(k>n, 2^n, b(n, [0$k^2])):
seq(seq(A(n, 1+d-n), n=0..d), d=0..11);

Mathematica

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 *)

Crossrefs

Columns k=1-6 give: A000012, A006253, A233289, A233291, A233294, A233424.

Diagonals include: A000079, A068156.

Sequence in context: A082386 A336521 A356093 * A028306 A111259 A304195

Adjacent sequences: A233305 A233306 A233307 * A233309 A233310 A233311

Keyword

nonn,tabl

Author

Alois P. Heinz, Dec 07 2013

Status

approved