A153445 -id:A153445 - cp's OEIS frontend

A055503 Take n points in general position in the plane; draw all the (infinite) straight lines joining them; sequence gives number of connected regions formed.

1, 1, 2, 7, 18, 41, 85, 162, 287, 478, 756, 1145, 1672, 2367, 3263, 4396, 5805, 7532, 9622, 12123, 15086, 18565, 22617, 27302, 32683, 38826, 45800, 53677, 62532, 72443, 83491, 95760, 109337, 124312, 140778, 158831, 178570, 200097, 223517, 248938, 276471

Offset: 0

N. J. A. Sloane, Jul 10 2000; Jul 03 2012

Jul 02 2012: Duane DeTemple points out that one could argue that a(1) should be 0, not 1, since if the single point is removed from the plane, the result is not simply connected (and then the formula given below applies for all n). However, the sequence as described by Comtet only specifies "connected", not "simply connected", so I prefer to have a(1)=1. - N. J. A. Sloane, Jul 03 2012

n points in general position determine "n choose 2" lines, so a(n) <= A000124(n(n-1)/2). If n > 3, the lines are not in general position and so a(n) < A000124(n(n-1)/2). - Jonathan Sondow, Dec 01 2015

			For n=2: draw three vertices forming a triangle and the three infinite straight lines joining them. There are a(3) = 7 connected regions.

L. Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, p. 72; and Problem 8, p. 74.

T. D. Noe, Table of n, a(n) for n = 0..1000
Olympiad Portal 74, Face-to-face rounds of the Chelyabinsk Region Open Olympiad 2008 (in Russian). See Mathematics 11 grade, problem 4.
Michal Opler, Pavel Valtr, and Tung Anh Vu, On the Arrangement of Hyperplanes Determined by n Points, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A000124, A263883. Subsequence of A177862.

3-dimensional equivalent: A153445.

A055503 := n->(1/8)*(n^4-6*n^3+23*n^2-26*n+8); [for n >1]

Join[{1,1},Table[(1/8)(n-1)(n^3-5n^2+18n-8),{n,2,80}]] (* Harvey P. Dale, May 06 2011 *)

a(n) = (1/8)*(n-1)*(n^3-5*n^2+18*n-8) for n>1.
For n>1: a(0)=2, a(1)=7, a(2)=18, a(3)=41, a(4)=85, a(n)=5a(n-1)- 10a(n-2)+ 10a(n-3)-5a(n-4)+a(n-5). [Harvey P. Dale, May 06 2011]
For n>1, G.f.: (-2+3x-3x^2-x^3)/(-1+x)^5. [Harvey P. Dale, May 06 2011]

a(1) changed from 0 to 1 by N. J. A. Sloane, Dec 07 2008

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.

Original entry on oeis.org

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

Nicolay Avilov, Jul 22 2023

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

			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.

Nicolay Avilov, Problem 1680. Tetrahedron and planes (in Russian).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000125, A153445, A248598.

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

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

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

cp's OEIS Frontend

A055503 Take n points in general position in the plane; draw all the (infinite) straight lines joining them; sequence gives number of connected regions formed.

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