A286442 Number of ways to tile an n X n X n triangular area with seven 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-28) of 1 X 1 X 1 tiles.

0, 57, 9233, 563287, 12649059, 152516103, 1211235921, 7147857411, 33812251267, 134823778299, 469266000129, 1462057867743, 4154650828483, 10922915001087, 26867398129329, 62381437357035, 137705497065315, 290721776912275, 589883390417697, 1155073034088999, 2190429436721571

Offset

5,2

Comments

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

For an analogous problem concerning square tiles, see A194788.

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 (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

Formula

a(n) = (n^14 -21*n^13 +2835*n^11 -13664*n^10 -147903*n^9 +1159368*n^8 +3480705*n^7 -44292941*n^6 -24613344*n^5 +908186412*n^4 -372748320*n^3 -9895978296*n^2 +5596762608*n +46620962640)/5040 for n>=8.

G.f.: x^6*(57 + 8378*x + 430777*x^2 + 5143284*x^3 + 17802143*x^4 + 7781860*x^5 - 20367093*x^6 - 406014*x^7 + 12253687*x^8 - 5320950*x^9 - 731329*x^10 + 627984*x^11 + 198177*x^12 - 135016*x^13 + 10557*x^14 - 198*x^15 + 976*x^16) / (1 - x)^15. - Colin Barker, May 16 2017

Example

There are 57 ways of tiling a triangular area of side 6 with 7 tiles of side 2 and an appropriate number (= 8) of tiles of side 1. See illustration in links section.

Prog

(PARI) concat(0, Vec(x^6*(57 + 8378*x + 430777*x^2 + 5143284*x^3 + 17802143*x^4 + 7781860*x^5 - 20367093*x^6 - 406014*x^7 + 12253687*x^8 - 5320950*x^9 - 731329*x^10 + 627984*x^11 + 198177*x^12 - 135016*x^13 + 10557*x^14 - 198*x^15 + 976*x^16) / (1 - x)^15 + O(x^30))) \\ Colin Barker, May 16 2017

Crossrefs

Cf. A194788, A286436, A286437, A286438, A286439, A286440, A286441.

Sequence in context: A218353 A127455 A263669 * A219077 A091749 A218425

Adjacent sequences: A286439 A286440 A286441 * A286443 A286444 A286445

Keyword

nonn,easy

Author

Heinrich Ludwig, May 15 2017

Status

approved