A289228 Number of ways to select 7 disjoint point triples from an n X n X n triangular point grid, each point triple forming a 2 X 2 X 2 triangle.
0, 2910, 404746, 12025068, 165279612, 1405048082, 8605979390, 41555851716, 167529980320, 586136559350, 1829074790082, 5193890370940, 13625393372916, 33410188057962, 77284672892438, 169909353488372, 357177283295160, 721559475338446, 1406717921047994, 2656028041092876
Offset: 6
Examples
There are 2910 ways to choose seven 2 X 2 X 2 triangles (aaa, ..., ggg) from a 7 X 7 X 7 point grid, for example:
a
a a
b . c
b b c c
d d . e e
f d . . e g
f f . . . g g
Note: aaa, ..., ggg are not distinguishable, they are denoted differently for a better perception of the 2 X 2 X 2 triangles only.
Links
- Heinrich Ludwig, Table of n, a(n) for n = 6..100
- Index entries for linear recurrences with constant coefficients, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).
Programs
-
PARI
concat(0, Vec(2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15 + O(x^40))) \\ Colin Barker, Jul 01 2017
Formula
a(n) = (n^14 -14*n^13 -182*n^12 +3416*n^11 +9072*n^10 -342062*n^9 +296688*n^8 +17893944*n^7 -48845153*n^6 -511039228*n^5 +2041220174*n^4 +7429535400*n^3 -37737333320*n^2 -41483946096*n +262680697440)/5040 for n>=7.
G.f.: 2*x^7*(1455 + 180548*x + 3129714*x^2 + 13038936*x^3 + 4149381*x^4 - 21524480*x^5 + 3658074*x^6 + 12791138*x^7 - 6864973*x^8 - 1299402*x^9 + 1667400*x^10 - 272962*x^11 + 37953*x^12 - 60178*x^13 + 16036*x^14) / (1 - x)^15. - Colin Barker, Jul 01 2017
Comments