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

0,3

Comments

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

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Author?, Title? (from Alexander Evnin, Dec 06 2008)

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).

Formula

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]

Example

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

Maple

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

Mathematica

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

Crossrefs

Cf. A000124, A263883. Subsequence of A177862.

3-dimensional equivalent: A153445.

Sequence in context: A054111 A295054 A192955 * A077802 A095151 A147611

Adjacent sequences: A055500 A055501 A055502 * A055504 A055505 A055506

Keyword

nonn,nice

Author

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

Extensions

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

Status

approved