A286439 Number of 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, 25, 747, 7459, 42983, 176373, 575775, 1595487, 3908979, 8701313, 17936083, 34713675, 63739327, 111921149, 189119943, 309074343, 490526475, 758575017, 1146284219, 1696579123, 2464458903, 3519561925, 4949117807, 6861323439, 9389181603, 12694842513, 16974490275

Offset

3,3

Comments

Rotations and reflections of tilings are counted. If they are to be ignored, see A286446. Tiles of the same size are not distinguishable.

For an analogous problem concerning square tiles, see A061997.

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 (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = (n^8 -12*n^7 +6*n^6 +432*n^5 -1279*n^4 -4692*n^3 +20592*n^2 +13320*n -91800)/24, for n>=5.

G.f.: x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9. - Colin Barker, May 12 2017

Example

There are 25 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 example in links section.

Prog

(PARI) concat(0, Vec(x^4*(1 + 16*x + 558*x^2 + 1552*x^3 + 770*x^4 - 1674*x^5 + 306*x^6 + 144*x^7 + 45*x^8 - 38*x^9) / (1 - x)^9 + O(x^60))) \\ Colin Barker, May 12 2017

Crossrefs

Cf. A286436, A286446, A286437, A286438, A286440, A286441, A286442, A061997.

Sequence in context: A266024 A217996 A246906 * A123835 A012835 A277584

Adjacent sequences: A286436 A286437 A286438 * A286440 A286441 A286442

Keyword

nonn,easy

Author

Heinrich Ludwig, May 11 2017

Status

approved