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.

%I A055503 #35 Apr 14 2025 12:06:06
%S A055503 1,1,2,7,18,41,85,162,287,478,756,1145,1672,2367,3263,4396,5805,7532,
%T A055503 9622,12123,15086,18565,22617,27302,32683,38826,45800,53677,62532,
%U A055503 72443,83491,95760,109337,124312,140778,158831,178570,200097,223517,248938,276471
%N 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.
%C A055503 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
%C A055503 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
%D A055503 L. Comtet, Advanced Combinatorics, Reidel, 1974, Problem 1, p. 72; and Problem 8, p. 74.
%H A055503 T. D. Noe, <a href="/A055503/b055503.txt">Table of n, a(n) for n = 0..1000</a>
%H A055503 Olympiad Portal 74, <a href="http://olymp74.ru/?razd=2&amp;page=event&amp;id=128">Face-to-face rounds of the Chelyabinsk Region Open Olympiad 2008</a> (in Russian). See <a href="http://olymp74.ru/files/view.php?id=9691f121716142771dba9cc2bf153a18fd65ee68edc48c1ea0d5a582d5eb2b13">Mathematics 11 grade</a>, problem 4.
%H A055503 Michal Opler, Pavel Valtr, and Tung Anh Vu, <a href="https://dccg.upc.edu/eurocg23/wp-content/uploads/2023/03/Session-7B-Talk-1.pdf">On the Arrangement of Hyperplanes Determined by n Points</a>, EuroCG (39th European Workshop on Computational Geometry, Barcelona, Spain 2023) Session 7B, Talk 1, Vol. 54, No. 6.
%H A055503 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).
%F A055503 a(n) = (1/8)*(n-1)*(n^3-5*n^2+18*n-8) for n>1.
%F A055503 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]
%F A055503 For n>1, G.f.: (-2+3x-3x^2-x^3)/(-1+x)^5. [_Harvey P. Dale_, May 06 2011]
%e A055503 For n=2: draw three vertices forming a triangle and the three infinite straight lines joining them. There are a(3) = 7 connected regions.
%p A055503 A055503 := n->(1/8)*(n^4-6*n^3+23*n^2-26*n+8); [for n >1]
%t A055503 Join[{1,1},Table[(1/8)(n-1)(n^3-5n^2+18n-8),{n,2,80}]] (* _Harvey P. Dale_, May 06 2011 *)
%Y A055503 Cf. A000124, A263883. Subsequence of A177862.
%Y A055503 3-dimensional equivalent: A153445.
%K A055503 nonn,nice
%O A055503 0,3
%A A055503 _N. J. A. Sloane_, Jul 10 2000; Jul 03 2012
%E A055503 a(1) changed from 0 to 1 by _N. J. A. Sloane_, Dec 07 2008