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

3,3

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.

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).

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A245986 A225810 A067196 * A048863 A363989 A278356

Adjacent sequences: A286443 A286444 A286445 * A286447 A286448 A286449

Keyword

nonn,easy

Author

Heinrich Ludwig, May 12 2017

Status

approved