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Search: id:A006066
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| A006066 |
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Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.
(Formerly M1334)
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+0
1
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OFFSET
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1,4
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COMMENT
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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]
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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 (njas(AT)research.att.com), 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.
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Gardner, Wheels, Life and Other Mathematical Amusements. Freeman, NY, 1983, p. 171.
Branko Gruenbaum, Convex Polytopes; 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.
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LINKS
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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
Alexandre Wajnberg, Illustration showing a(10) >= 25 [A different construction from Gruenbaum's]
Eric Weisstein's World of Mathematics, Kobon Triangle
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FORMULA
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An upper bound on this sequence is given by A032765.
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EXAMPLE
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a(17) = 85 because the a configuration with 85 exists meeting the upper bound.
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CROSSREFS
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Sequence in context: A157001 A134640 A032616 this_sequence A084935 A062409 A089781
Adjacent sequences: A006063 A006064 A006065 this_sequence A006067 A006068 A006069
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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a(15) = 65 found by T. Suzuki on Oct 02, 2005. - Eric Weisstein (eric(AT)weisstein.com), Oct 04, 2005.
Gruenbaum reference from Anthony Labarre, Dec 19 2005
Additional links to Japanese web sites from Alexandre Wajnberg (alexandre.wajnberg(AT)skynet.be), Dec 29 2005 and Anthony Labarre (alabarre(AT)ulb.ac.be), 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".
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