A286444 Number of non-equivalent ways to tile an n X n X n triangular area with two 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-8) of 1 X 1 X 1 tiles.

0, 3, 10, 32, 70, 143, 252, 424, 660, 995, 1430, 2008, 2730, 3647, 4760, 6128, 7752, 9699, 11970, 14640, 17710, 21263, 25300, 29912, 35100, 40963, 47502, 54824, 62930, 71935, 81840, 92768, 104720, 117827, 132090, 147648, 164502, 182799, 202540, 223880, 246820, 271523

Offset

3,2

Comments

Rotations and reflections of tilings are not counted. If they are to be counted, see A286437. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A279111.

Links

Heinrich Ludwig, Table of n, a(n) for n = 3..100

Heinrich Ludwig, Illustration of tiling a 4X4X4 area

Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Formula

a(n) = (n^4 -6*n^3 +11*n^2 -12)/12 + IF(MOD(n, 2) = 1, -n +2)/2.

G.f.: x^4*(3 + x + 5*x^2 - x^3) / ((1 - x)^5*(1 + x)^2). - Colin Barker, May 12 2017

Example

There are 3 non-equivalent ways of tiling a triangular area of side 4 with two tiles of side 2 and an appropriate number (= 8) of tiles of side 1. See example in links section.

Prog

(PARI) concat(0, Vec(x^4*(3 + x + 5*x^2 - x^3) / ((1 - x)^5*(1 + x)^2) + O(x^30))) \\ Colin Barker, May 12 2017

Crossrefs

Cf. A279111, A286437, A286443, A286445, A286446.

Sequence in context: A034016 A001403 A072136 * A080406 A036682 A104270

Adjacent sequences: A286441 A286442 A286443 * A286445 A286446 A286447

Keyword

nonn,easy

Author

Heinrich Ludwig, May 12 2017

Status

approved