A286437 Number of 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, 9, 48, 153, 372, 765, 1404, 2373, 3768, 5697, 8280, 11649, 15948, 21333, 27972, 36045, 45744, 57273, 70848, 86697, 105060, 126189, 150348, 177813, 208872, 243825, 282984, 326673, 375228, 428997, 488340, 553629, 625248, 703593, 789072, 882105, 983124, 1092573

Offset

3,2

Comments

Rotations and reflections of tilings are counted. If they are to be ignored, see A286444. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A061995.

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 (5,-10,10,-5,1).

Formula

a(n) = (n^4 - 6*n^3 + 5*n^2 + 30*n - 54)/2, n>=3.

From Colin Barker, May 12 2017: (Start)

G.f.: 3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.

(End)

Example

There are 9 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(3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5 + O(x^60))) \\ Colin Barker, May 12 2017

Crossrefs

Cf. A286436, A286444, A286438, A286439, A286440, A286441, A286442, A061995.

Sequence in context: A159525 A173895 A341757 * A212107 A073979 A018984

Adjacent sequences: A286434 A286435 A286436 * A286438 A286439 A286440

Keyword

nonn,easy

Author

Heinrich Ludwig, May 10 2017

Status

approved