

A006066


Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.
(Formerly M1334)


2




OFFSET

1,4


COMMENTS

The known values a = a(n) and upper bounds U (usually A032765(n)) with name of discoverer of the arrangement when known are as follows:
n a U [Found by]

1 0 0
2 0 0
3 1 1
4 2 2
5 5 5
6 7 7
7 11 11
8 15 16
9 21 21
10 25? 26 [Grünbaum]
11 32? 33 [See link below]
12 ? 40
13 47 47 [Kabanovitch]
14 >= 53 56 [Bader]
15 65 65 [Suzuki]
16 >=72 74 [Bader]
17 85 85 [Bader]
18 >= 93 96 [Bader]
19 >= 104 107 [Bader]
20 >= 115 120 [Bader]
21 >= 130 133 [Bader]
22 ? 146
23 ? 161
24 ? 176
25 ? 191
26 ? 208
27 ? 225
28 ? 242
29 ? 261
30 ? 280
31 ? 299
32 ? 320
Ed Pegg's web page gives the upper bound for a(6) as 8. But by considering all possible arrangements of 6 lines  the sixth term of A048872  one can see that 8 is impossible.  N. J. A. Sloane, Nov 11 2007
Although they are somewhat similar, this sequence is strictly different from A084935, since A084935(12) = 48 exceeds the upper bound on a(12) from A032765.  Floor van Lamoen, Nov 16 2005
The name is sometimes incorrectly entered as "Kodon" triangles.


REFERENCES

M. Gardner, Wheels, Life and Other Mathematical Amusements. Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura.
Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; p. 400 shows that a(10) >= 25.
Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, by the Russian puzzle club Diogen), pp. 12, June 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..9.
J. Bader, Kobon Triangles
J. Bader, Kobon Triangles [Cached copy, with permission, pdf format]
J. Bader, Illustration showing a(17)=85, Nov 28 2007.
J. Bader, Illustration showing a(17)=85, Nov 28 2007. [Cached copy, with permission]
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007
Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007 [Cached copy, with permission]
S. Honma, Title? (A related site)
S. Honma, Title? (A related site)
S. Honma, Illustration showing a(11)>=32
S. Honma, Title? (A related site)
S. Honma, Title? (A related site)
S. Honma, Title? (A related site)
Ed Pegg, Jr., Kobon triangles
Ed Pegg, Jr., Kobon Triangles [Cached copy, with permission, pdf format]
N. J. A. Sloane, Illustration for a(5) = 5 (a pentagram)
Alexandre Wajnberg, Illustration showing a(10) >= 25 [A different construction from Grünbaum's]
Eric Weisstein's World of Mathematics, Kobon Triangle


FORMULA

An upper bound on this sequence is given by A032765.
For any odd n > 1, if n == 1 (mod 6), a(n) <= (n^2  (2n + 2))/3; in other odd cases, a(n) <= (n^2  2n)/3. For any even n > 0, if n == 4 (mod 6), a(n) <= (n^2  (2n + 2))/3, otherwise a(n) <= (n^2  2n)/3.  Sergey Pavlov, Feb 11 2017


EXAMPLE

a(17) = 85 because the a configuration with 85 exists meeting the upper bound.


CROSSREFS

Sequence in context: A216094 A184857 A032616 * A084935 A239072 A217302
Adjacent sequences: A006063 A006064 A006065 * A006067 A006068 A006069


KEYWORD

nonn,hard,more,changed


AUTHOR

N. J. A. Sloane


EXTENSIONS

a(15) = 65 found by Toshitaka Suzuki on Oct 02 2005.  Eric W. Weisstein, Oct 04 2005
Grünbaum reference from Anthony Labarre, Dec 19 2005
Additional links to Japanese web sites from Alexandre Wajnberg, Dec 29 2005 and Anthony Labarre, Dec 30 2005
A perfect solution for 13 lines was found in 1999 by Kabanovitch.  Ed Pegg Jr, Feb 08 2006
Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), Dec 06 2007, who says "Acknowledgments and dedication to Corinne Thomet".


STATUS

approved



