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A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.
(Formerly M1334)
1
0, 0, 1, 2, 5, 7, 11, 15, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

The known values a = a(n) and upperbounds 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 [Gruenbaum]

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 en Lyanne van Lamoen (fvanlamoen(AT)planet.nl), 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 Gruenbaum, 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. 1-2, 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, Illustration showing a(17)=85, Nov 28 2007.

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

N. J. A. Sloane, Illustration for a(5) = 5

Alexandre Wajnberg, Illustration showing a(10) >= 25 [A different construction from Gruenbaum's]

Eric Weisstein's World of Mathematics, Kobon Triangle

FORMULA

An upper bound on this sequence is given by A032765.

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

AUTHOR

N. J. A. Sloane.

EXTENSIONS

a(15) = 65 found by Toshitaka Suzuki on Oct 02, 2005. - Eric W. Weisstein, Oct 04, 2005.

Gruenbaum 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

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Last modified October 20 06:03 EDT 2014. Contains 248329 sequences.