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

Author

Heinrich Ludwig, May 10 2017

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

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

Programs

PARI

concat(0, Vec(3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5 + O(x^60))) \\ Colin Barker, May 12 2017

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)