A286436 Irregular triangle read by rows: T(n, k) = number of 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, 3, 1, 7, 9, 4, 1, 1, 13, 48, 63, 25, 1, 21, 153, 494, 747, 546, 219, 57, 9, 1, 1, 31, 372, 2247, 7459, 14064, 15160, 9233, 3069, 480, 14, 1, 43, 765, 7396, 42983, 157248, 369787, 563287, 556932, 358974, 153520, 45282, 9634, 1529, 186, 16, 1, 1, 57, 1404

Offset

1,5

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 counted. If they are to be ignored, see A286443. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A193580.

Links

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

Example

The triangle begins with T(1, 0):
1;
1,  1;
1,  3;
1,  7,   9,   4,   1;
1, 13,  48,  63,  25;
1, 21, 153, 494, 747, 546, 219, 57, 9, 1;
T(4, 3) = 4 because there are 4 ways to tile an area of size 4X4X4 with 3 tiles of size 2X2X2 and fill up the rest with tiles of size 1X1X1.

Crossrefs

Cf. A193580, A286443, A286437, A286438, A286439, A286440, A286441, A286442.

Sequence in context: A329943 A011207 A087129 * A011308 A347840 A033465

Adjacent sequences: A286433 A286434 A286435 * A286437 A286438 A286439

Keyword

nonn,tabf

Author

Heinrich Ludwig, May 16 2017

Status

approved