A216172 Number of all possible tetrahedra of any size, having reverse orientation to the original regular tetrahedron, formed when intersecting the latter by planes parallel to its sides and dividing its edges into n equal parts.

0, 0, 1, 4, 10, 21, 39, 66, 105, 159, 231, 325, 445, 595, 780, 1005, 1275, 1596, 1974, 2415, 2926, 3514, 4186, 4950, 5814, 6786, 7875, 9090, 10440, 11935, 13585, 15400, 17391, 19569, 21945, 24531, 27339, 30381, 33670, 37219, 41041, 45150, 49560, 54285, 59340

Offset

1,4

Comments

The number of all possible tetrahedra of any size, having the same orientation as the original regular tetrahedron is given by A000332(n+3).

Create a sequence wherein the sum of three consecutive numbers is a triangular number: 0,0,0,1,2,3,5,7...; then find the partial sums of this sequence: 0,0,0,1,3,6,11,18...; then take the partial sums of this sequence: 0,0,0,1,4,10,21,39,66... and after dropping the first two zeros, you get this sequence. - J. M. Bergot, Apr 14 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) = (1/72)*(-6*n -5*n^2 +2*n^3 +n^4 +4 -4*(-1)^(n mod 3)).

G.f.: x^3/((1-x)^5*(1+x+x^2)). - Bruno Berselli, Sep 11 2012

a(3*n-1) = A000217(A115067(n)); a(3*n) = A000217(A095794(n)); a(3*n+1) = A000217(A143208(n+2)) + A000217(n). - J. M. Bergot, Apr 14 2016

E.g.f.: (1/216)*(8 - 24*x + 24*x^2 + 24*x^3 + 3*x^4)*exp(x) - (1/27)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2))*exp(-x/2). - Ilya Gutkovskiy, Apr 14 2016

Example

For n=9 the numbers of the reversely oriented tetrahedra, starting from the smallest size, are A000292(7)=84, A000292(4)=20, and A000292(1)=1, the sum being a(9)=105.

Mathematica

nnn = 100; Tev[n_] := (n - 2) (n - 1) n/6; Table[Sum[Tev[n - nn], {nn, 0, n - 1, 3}], {n, nnn}]
Table[(1/72) (-6 n - 5 n^2 + 2 n^3 + n^4 + 4 - 4 (-1)^Mod[n, 3]), {n, 50}]
CoefficientList[Series[x^2 / ((1 - x)^5*(1 + x + x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 12 2012 *)
LinearRecurrence[{4, -6, 5, -5, 6, -4, 1}, {0, 0, 1, 4, 10, 21, 39}, 50] (* Harvey P. Dale, Feb 18 2018 *)

Prog

(Magma) I:=[0, 0, 1, 4, 10, 21, 39]; [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
(PARI) a(n)=(n^4+2*n^3-5*n^2-6*n+4-4*(-1)^(n%3))/72 \\ Charles R Greathouse IV, Sep 12 2012

Crossrefs

Cf. A000292, A000332, A216173, A216175.

Sequence in context: A253687 A253688 A049480 * A055908 A325951 A023538

Adjacent sequences: A216169 A216170 A216171 * A216173 A216174 A216175

Keyword

nonn,easy

Author

V.J. Pohjola, Sep 03 2012

Status

approved