login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 5
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 (list; graph; refs; listen; history; text; internal format)
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)

FORMULA

(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.

Sequence in context: A051743 A054111 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 28 10:34 EDT 2017. Contains 287240 sequences.