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.
1, 6, 20, 50, 104, 193, 329, 526, 800, 1169, 1652, 2271, 3049, 4011, 5184, 6597, 8280, 10266, 12589, 15285, 18392, 21950, 26000, 30586, 35753, 41548, 48020, 55220, 63200, 72015, 81721, 92376, 104040, 116775, 130644, 145713, 162049, 179721, 198800, 219359
Offset: 1
Examples
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).
Links
- Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).
Programs
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Mathematica
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}]
Formula
a(n) = (1/288)*(7+9*(-1)^n-16*(-1)^(n mod 3)+24*n+124*n^2+104*n^3+22*n^4).
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