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

1,2

Comments

This design neither includes planes passing through the vertex of the tetrahedron parallel to the opposite face, nor planes that extend the faces.

Links

Table of n, a(n) for n=1..44.

Nicolay Avilov, Problem 1680. Tetrahedron and planes (in Russian).

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

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

Example

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.

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 15, 65, 174}, 42] (* Robert P. P. McKone, Aug 25 2023 *)

Crossrefs

Cf. A000125, A153445, A248598.

Sequence in context: A096905 A147857 A147858 * A005917 A218216 A027455

Adjacent sequences: A364398 A364399 A364400 * A364402 A364403 A364404

Keyword

nonn,easy

Author

Nicolay Avilov, 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

Status

approved