cp's OEIS Frontend

A216175 Number of all polyhedra (tetrahedra of any orientation and octahedra) of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts.

%I A216175 #20 Feb 17 2024 08:14:16
%S A216175 1,6,20,50,104,193,329,526,800,1169,1652,2271,3049,4011,5184,6597,
%T A216175 8280,10266,12589,15285,18392,21950,26000,30586,35753,41548,48020,
%U A216175 55220,63200,72015,81721,92376,104040,116775,130644,145713,162049,179721,198800,219359
%N A216175 Number of all polyhedra (tetrahedra of any orientation and octahedra) of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts.
%H A216175 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,0,1,2,-3,1).
%F A216175 a(n) = (1/288)*(7+9*(-1)^n-16*(-1)^(n mod 3)+24*n+124*n^2+104*n^3+22*n^4).
%F A216175 G.f.: x*(1+3*x+4*x^2+3*x^3)/((1+x)*(1+x+x^2)*(1-x)^5). - _Bruno Berselli_, Sep 11 2012
%e A216175 For n=3, the number of tetrahedra of any orientation and size is t(3)+t(1)=15+1=16 and the number of octahedra of any size is t(2)=4 the total number being a(n)=20, where t(n) denotes the tetrahedral number A000292(n).
%t A216175 Table[(1/288) (7 + 9 (-1)^n - 16 (-1)^Mod[n, 3] + 24 n + 124 n^2 + 104 n^3 + 22 n^4), {n, 50}]
%Y A216175 Cf. A000292, A216172, A216173.
%K A216175 nonn,easy
%O A216175 1,2
%A A216175 _V.J. Pohjola_, Sep 03 2012