login
A286437
Number of ways to tile an n X n X n triangular area with two 2 X 2 X 2 triangular tiles and an appropriate number (= n^2-8) of 1 X 1 X 1 tiles.
8
0, 9, 48, 153, 372, 765, 1404, 2373, 3768, 5697, 8280, 11649, 15948, 21333, 27972, 36045, 45744, 57273, 70848, 86697, 105060, 126189, 150348, 177813, 208872, 243825, 282984, 326673, 375228, 428997, 488340, 553629, 625248, 703593, 789072, 882105, 983124, 1092573
OFFSET
3,2
COMMENTS
Rotations and reflections of tilings are counted. If they are to be ignored, see A286444. Tiles of the same size are indistinguishable.
For an analogous problem concerning square tiles, see A061995.
FORMULA
a(n) = (n^4 - 6*n^3 + 5*n^2 + 30*n - 54)/2, n>=3.
From Colin Barker, May 12 2017: (Start)
G.f.: 3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>7.
(End)
EXAMPLE
There are 9 ways of tiling a triangular area of side 4 with two tiles of side 2 and an appropriate number (= 8) of tiles of side 1. See example in links section.
PROG
(PARI) concat(0, Vec(3*x^4*(3 + x + x^2 - x^3) / (1 - x)^5 + O(x^60))) \\ Colin Barker, May 12 2017
KEYWORD
nonn,easy
AUTHOR
Heinrich Ludwig, May 10 2017
STATUS
approved