A243608 Number T(n,k) of ways k L-tiles can be placed on an n X n square; triangle T(n,k), n>=0, 0<=k<=A229093(n), read by rows.

1, 1, 1, 1, 1, 4, 1, 1, 9, 20, 11, 1, 1, 16, 87, 196, 176, 46, 2, 1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2, 1, 36, 545, 4544, 22969, 73098, 147502, 185744, 140288, 59140, 12313, 1046, 26, 1, 49, 1056, 13215, 106819, 587149, 2251309, 6082000, 11562155

Offset: 0

Author

Alois P. Heinz, Jun 07 2014

Keywords

• nonn
• tabf

Comments

An L-tile is a 2 X 2 square with the upper right 1 X 1 subsquare removed and no rotations are allowed.

Examples

			T(3,1) = 4:
  ._____.   ._____.   ._____.   ._____.
  | |_|_|   |_|_|_|   |_| |_|   |_|_|_|
  |___|_|   | |_|_|   |_|___|   |_| |_|
  |_|_|_|   |___|_|   |_|_|_|   |_|___|
T(4,4) = 1:
  ._______.
  | |_| |_|
  |___|___|
  | |_| |_|
  |___|___|
T(5,6) = 2:
  ._________.   ._________.
  | |_|_| |_|   |_| |_| |_|
  |___| |___|   | |___|___|
  |_| |___|_|   |___|_| |_|
  | |___| |_|   | |_| |___|
  |___|_|___|   |___|___|_| .
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  4,   1;
  1,  9,  20,   11,    1;
  1, 16,  87,  196,  176,   46,    2;
  1, 25, 244, 1195, 3145, 4431, 3161, 1007, 111, 2;
		

Links

• Alois P. Heinz, Rows n = 0..21, flattened

Crossrefs

Columns k=0-6 give: A000012, A000290(n-1) for n>0, A243645, A243646, A243647, A243648, A243649.

Row sums give main diagonal of A226444 or A066864(n-1) for n>0.

Programs

Maple

b:= proc(n, l) option remember; local k;
      if n<2 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(
         b(n, subsop(k=1, l))+ `if`(n>1 and k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..10);

Mathematica

b[n_, l_] := b[n, l] = Module[{k}, Which[n<2, 1, Min[l]>0, b[n-1, l-1], True, For[k = 1, l[[k]] > 0, k++]; Expand[b[n, ReplacePart[l, k -> 1]] + If[n>1 && k 2, k+1 -> 1}]], 0]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, Table[0, {n}]]];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 12 2017, translated from Maple *)