A216173 Number of all possible tetrahedra of any size and orientation, formed when intersecting the original regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts.

1, 5, 16, 39, 80, 147, 249, 396, 600, 874, 1232, 1690, 2265, 2975, 3840, 4881, 6120, 7581, 9289, 11270, 13552, 16164, 19136, 22500, 26289, 30537, 35280, 40555, 46400, 52855, 59961, 67760, 76296, 85614, 95760, 106782, 118729, 131651, 145600, 160629, 176792

Offset

1,2

Comments

A269747 is a very similar sequence with a much simpler definition (but which does not appear to have a simple formula). The difference between these two sequences is unclear. - N. J. A. Sloane, Mar 30 2016

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Formula

a(n) = (1/18)*(1 - (-1)^(n mod 3) + 3*n + 7*n^2 + 5*n^3 + n^4).

a(n) = A000332(n+3) + A216172(n).

G.f.: x*(1+x+2*x^2)/((1-x)^5*(1+x+x^2)). - Bruno Berselli, Sep 11 2012

Example

For n=9 the number of the tetrahedra of any orientation (those with the same orientation as the original one plus those with reverse orientation) is the total of the number of each size (counted from the smallest size upwards): t(9)+t(7)=165+84=249, t(8)+t(4)=120+20=140, t(7)+t(1)=84+1=85, t(6)=56, t(5)=35, t(4)=20, t(3)=10, t(2)=4 and t(1)=1, the total being a(9)=600, where t(n) denotes the tetrahedral number A000292(n).

Mathematica

Table[(1/18) (1 - (-1)^Mod[n, 3] + 3 n + 7 n^2 + 5 n^3 + n^4), {n, 50}]
CoefficientList[Series[(1 + x + 2 x^2)/((1 - x)^5 (1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2012 *)

Prog

(Magma) I:=[1, 5, 16, 39, 80, 147, 249]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..50]]; // Vincenzo Librandi, Sep 12 2012

Crossrefs

Cf. A000292, A000332, A216172, A216175.

A269747 is a very similar sequence.

Sequence in context: A082199 A082190 A155965 * A269747 A011863 A027085

Adjacent sequences: A216170 A216171 A216172 * A216174 A216175 A216176

Keyword

nonn,easy

Author

V.J. Pohjola, Sep 03 2012

Status

approved