A286443 Irregular triangle read by rows: T(n, k) = number of non-equivalent ways to tile an n X n X n triangular area with k 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-4*k) of 1 X 1 X 1 tiles.

1, 1, 1, 1, 1, 1, 3, 3, 2, 1, 1, 4, 10, 14, 6, 1, 6, 32, 97, 142, 105, 46, 14, 3, 1, 1, 8, 70, 398, 1280, 2386, 2574, 1569, 524, 87, 3, 1, 11, 143, 1290, 7301, 26471, 62067, 94423, 93358, 60287, 25881, 7697, 1678, 281, 40, 5, 1, 1, 13, 252, 3366, 29603, 176591, 728868

Offset: 1

Author

Heinrich Ludwig, May 16 2017

Keywords

• nonn
• tabf

Comments

The triangle T(n, k) is irregularly shaped: For n >= 4: 0 <= k <= n^2/4 if n is even, 0 <= k <= (n^2 -9)/4 if n is odd. First row corresponds to n = 1.

Rotations and reflections of tilings are not counted. If they are to be counted, see A286436. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A236679.

Examples

			The triangle begins with T(1, 0)
   1;
   1,    1;
   1,    1;
   1,    3,    3,    2,    1;
   1,    4,   10,   14,    6;
   1,    6,   32,   97,  142,  105,   46,   14,    3,    1;
   1,    8,   70,  398, 1280, 2386, 2574, 1569,  524,   87,    3;
T(4, 3) = 2 because there are 2 non-equivalent ways to tile an area of size 4X4X4 with 3 tiles of size 2X2X2 and fill up the rest with tiles of size 1X1X1.
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..140

Crossrefs

Cf. A236679, A286436, A286444, A286445, A286446.