A364401 a(n) is the number of regions into which three-dimensional Euclidean space is divided by n-1 planes parallel to each face of a regular tetrahedron with edge length n, dividing the edges into unit segments.
1, 15, 65, 174, 365, 661, 1085, 1660, 2409, 3355, 4521, 5930, 7605, 9569, 11845, 14456, 17425, 20775, 24529, 28710, 33341, 38445, 44045, 50164, 56825, 64051, 71865, 80290, 89349, 99065, 109461, 120560, 132385, 144959, 158305, 172446, 187405, 203205, 219869, 237420, 255881, 275275, 295625, 316954
Offset: 1
Examples
a(1) = 1, there are no planes and all space is one part; a(2) = 1 + 4 + 4 + 6 = 15 because in this case there are four planes defining a tetrahedron. These four planes divide the space into 15 parts, namely: 1 part - the inside of the tetrahedron; 4 parts are adjacent to the faces of the tetrahedron; 4 parts are adjacent to the vertices of the tetrahedron; 6 parts are adjacent to the edges of the tetrahedron; a(3) = (23*27 - 30*9 + 13*3)/6 = 65.
Links
- Nicolay Avilov, Problem 1680. Tetrahedron and planes (in Russian).
- Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
Programs
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Mathematica
LinearRecurrence[{4, -6, 4, -1}, {1, 15, 65, 174}, 42] (* Robert P. P. McKone, Aug 25 2023 *)
Formula
a(n) = (23*n^3 - 30*n^2 + 13*n)/6 [from Anatoly Kazmerchuk].
G.f.: x*(1 + 11*x + 11*x^2)/(1 - x)^4. - Stefano Spezia, Jul 22 2023
Extensions
a(1) = 1 inserted by Nicolay Avilov, Oct 19 2023
Name corrected by Talmon Silver, Oct 29 2023
a(44) = 316954 added by Talmon Silver, Dec 01 2023
Comments