A286441 Number of ways to tile an n X n X n triangular area with six 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-24) of 1 X 1 X 1 tiles.

0, 219, 15160, 369787, 4366982, 32450843, 175628996, 755759531, 2734928266, 8643796747, 24503068784, 63522668395, 152816062222, 345005930315, 737473609532, 1503178571195, 2938515130514, 5535661080283, 10089397100584, 17851538034587, 30750030827926, 51694565135803

Offset: 5

Author

Heinrich Ludwig, May 12 2017

Keywords

• nonn
• easy

Comments

Rotations and reflections of tilings are counted. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A172158.

Examples

			There are 219 ways of tiling a triangular area of side 6 with 6 tiles of side 2 and an appropriate number (= 12) of tiles of side 1. See illustration in links section.
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 5..100
• Heinrich Ludwig, Illustration of tiling a 6X6X6 area
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A172158, A286436, A286437, A286438, A286439, A286440, A286442.

Programs

PARI

concat(0, Vec( x^6*(219 + 12313*x + 189789*x^2 + 679597*x^3 + 344288*x^4 - 808902*x^5 + 54074*x^6 + 289970*x^7 - 51453*x^8 - 71891*x^9 + 27785*x^10 - 255*x^11 + 98*x^12 - 352*x^13) / (1 - x)^13 + O(x^60))) \\ Colin Barker, May 13 2017

Formula

a(n) = (n^12 - 18*n^11 + 3*n^10 + 1710*n^9 - 7175*n^8 - 60078*n^7 + 401649*n^6 + 884466*n^5 - 9521846*n^4 - 3238224*n^3 + 107453448*n^2 - 25651296*n - 483140880)/720 for n >= 7.

G.f.: x^6*(219 + 12313*x + 189789*x^2 + 679597*x^3 + 344288*x^4 - 808902*x^5 + 54074*x^6 + 289970*x^7 - 51453*x^8 - 71891*x^9 + 27785*x^10 - 255*x^11 + 98*x^12 - 352*x^13) / (1 - x)^13. - Colin Barker, May 13 2017