A286446 Number of non-equivalent ways to tile an n X n X n triangular area with four 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-16) of 1 X 1 X 1 tiles.

0, 1, 6, 142, 1280, 7301, 29603, 96485, 266636, 652908, 1452054, 2992513, 5789499, 10629381, 18660890, 31530854, 51525116, 81772345, 126449707, 191075297, 282794784, 410784700, 586640186, 824912741, 1143620051, 1564946921, 2115898646, 2829194838, 3744093216, 4907506597

Offset: 3

Author

Heinrich Ludwig, May 12 2017

Keywords

• nonn
• easy

Comments

Rotations and reflections of tilings are not counted. If they are to be counted, see A286439. Tiles of the same size are indistinguishable.

For an analogous problem concerning square tiles, see A279113.

Examples

			There are 6 non-equivalent ways of tiling a triangular area of side 5 with 4 tiles of side 2 and an appropriate number (= 9) of tiles of side 1. See illustration in links section.
		

Links

• Heinrich Ludwig, Table of n, a(n) for n = 3..100
• Heinrich Ludwig, Illustration of tiling a 5X5X5 area
• Index entries for linear recurrences with constant coefficients, signature (2,3,-5,-8,3,19,4,-24,-15,15,24,-4,-19,-3,8,5,-3,-2,1).

Crossrefs

Cf. A279113, A286439, A286443, A286444, A286445.

Programs

PARI

concat(0, Vec(x^4*(1 + 4*x + 127*x^2 + 983*x^3 + 4353*x^4 + 11916*x^5 + 22875*x^6 + 31058*x^7 + 30066*x^8 + 18947*x^9 + 5576*x^10 - 2441*x^11 - 3003*x^12 - 698*x^13 + 707*x^14 + 536*x^15 + 71*x^16 - 73*x^17 - 37*x^18 - 8*x^19) / ((1 - x)^9*(1 + x)^4*(1 + x + x^2)^3) + O(x^60))) \\ Colin Barker, May 12 2017

Formula

a(n) = (n^8 -12*n^7 +6*n^6 +432*n^5 -1249*n^4 -5028*n^3 +21820*n^2 +12384*n -94000)/144 + IF(MOD(n, 2) = 1, -8*n^3 +72*n^2 -208*n +189)/24 + IF(MOD(n, 3) = 0, -n^2 +3*n +7)/9 for n >= 5.

G.f.: x^4*(1 + 4*x + 127*x^2 + 983*x^3 + 4353*x^4 + 11916*x^5 + 22875*x^6 + 31058*x^7 + 30066*x^8 + 18947*x^9 + 5576*x^10 - 2441*x^11 - 3003*x^12 - 698*x^13 + 707*x^14 + 536*x^15 + 71*x^16 - 73*x^17 - 37*x^18 - 8*x^19) / ((1 - x)^9*(1 + x)^4*(1 + x + x^2)^3). - Colin Barker, May 12 2017