A048872 -id:A048872 - cp's OEIS frontend

A250001 Number of arrangements of n circles in the affine plane.

1, 1, 3, 14, 173, 16951

Offset: 0

Jon Wild, May 16 2014

Two circles are either disjoint or meet in two points. Tangential contacts are not allowed. A point belongs to exactly one or two circles. Three circles may not meet at a point. The circles may have different radii.
This is in the affine plane, rather than the projective plane.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all circles circular (although the radii may be continuously changed), without changing the multiplicity of intersection points, and without a circle passing through an intersection point. Turning the whole configuration over is allowed.
Several variations are possible:
- straight lines instead of circles (see A241600).
- straight lines in general position (see A090338).
- curved lines in general position (see A090339).
- allow circles to meet tangentially but without multiple intersection points (begins 1, 5, ...); more terms are needed.
- again use circles, but allow multiple intersection points (also begins 1, 5, ...); more terms are needed.
- use ellipses rather than circles.
- a question from Walter D. Wallis: what if the circles must all have the same radius?
a(1)-a(5) computed by Jon Wild.
a(n) >= A000081(n+1) - Benoit Jubin, Dec 21 2014. More precisely, there are A000081(n+1) ways to arrange n circles if no two of them meet. - N. J. A. Sloane, May 16 2017
From Daniel Forgues, Aug 08-09 2015: (Start)
A representation for the diagrams in a250001.jpg (in the same order):
  a(1) =  1: {{2}};
  a(2) =  3: {{2, 3}, {2, 4}, {4, 6}};
  a(3) = 14: {{2, 4, 8}, {2, 3, 6}, {2, 3, 4}, {2, 3, 5}, {4, 6, 5},
    {4, 6, 15}, {2, 6, 9}, {4, 6, 12}, {2, 8, 12}, {30, 42, 70},
    {?, ?, ?}, {?, ?, ?}, {15, 21, 35}, {?, ?, ?}}.
In lexicographic order:
  a(3) = 14: {{2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 8}, {2, 6, 9},
    {2, 8, 12}, {4, 6, 5}, {4, 6, 12}, {4, 6, 15}, {15, 21, 35},
    {30, 42, 70}, {?, ?, ?}, {?, ?, ?}, {?, ?, ?}}.
The smallest integers greater than 1 are used for the representation:
  (p_1)^(a_1)*...*(p_m)^(a_m), where
  0 <= a_i <= n, for 1 <= i <= m;
  (a_1)+...+(a_m) > 0.
Could the Venn diagram interpretation (of the k-wise, 1 <= k <= n, common divisors of k numbers from each subset) reveal a pattern?
Does this representation work for more complex diagrams? (End)
Once you get to n=5, geometric concerns mean that not all topologically-conceivable arrangements are actually circle-drawable.  My program enumerated 16977 conceivable arrangements of 5 pseudo-circles, and Christopher Jones and I together have figured out how to show that 26 of these arrangements are not actually circle-drawable. So it seems that a(5) = 16951. This entry will be updated soon, and there will be a new sequence for the number of topologically-conceivable arrangements. - Jon Wild, Aug 25 2016 [The counts in this comment were amended by Jon Wild on Aug 30 2016. I apologize for taking so long to make the corrections here. - N. J. A. Sloane, Jun 11 2017]
a(n) <= 7*13^(binomial(n,3) + binomial(n,2) + 3n - 1) is a (loose) upper bound, see Reddit link. I believe XkF21WNJ's reply shaves off a factor of 13^3 from this bound for all n > 1. - Linus Hamilton, Apr 14 2019
A good upper bound for a(6) is given in sequence A288559, which counts the arrangements of pseudo-circles, i.e. the topologically conceivable arrangements mentioned above, which are not all necessarily realizable with true circles. The number of arrangements of 6 pseudo-circles was found by Andrii Shportko and Jon Wild to be 17,552,169. - Jon Wild, Jun 03 2025
In A288559, a(5) included 26 non-circularizable pseudocircle arrangements, which generated in turn 132,546 6-pseudocircle descendants. These descendants must be excluded from A250001, which means that a tighter upper bound for A250001(6) is 17,419,623. - Andrii Shportko, Jun 06 2025

			a(2) = 3, because two circles can either be next to each other, overlap with two intersection points, or one may be located within the other (of larger radius). (As per the first comment, the limiting case where they touch in one point is [somewhat arbitrarily] excluded. This would add two more independent configurations, where one touched the other "from inside" or "from outside".) - _M. F. Hasler_, May 03 2025

Jon Wild, Posting to Sequence Fans Mailing List, May 15 2014.

Mohammad Arab, Creative proofs in combinations, arXiv:2112.08020 [math.CO], 2021-2022.
Andrew Cook and Luca Viganò, A Game Of Drones: Extending the Dolev-Yao Attacker Model With Movement, Proceedings of the 6th Workshop on Hot Issues in Security Principles and Trust (HotSpot 2020): Affiliated with Euro S&P 2020, IEEE Computer Science Press, Genova, Italy (2020).
Linus Hamilton, How many ways can circles overlap? - Numberphile, Reddit.
R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016. [Not directly related, but on a similar subject. - _N. J. A. Sloane_, Jan 20 2017]
N. J. A. Sloane, Illustration of a(2)=3 and a(3)=14
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, pp. 9, 21.
N. J. A. Sloane and Brady Haran, How many ways can circles overlap?, Numberphile video (2019)
Jon Wild, Illustrations of the 173 configurations of four circles
Jon Wild, Illustrations of the 112 connected configurations of four circles (Computer-generated svg file. To see it, save file, open it with a program - such as Chrome - that can handle svg files.)
Jon Wild, Figure showing relationship between A250001, A275923, A275924, and A288554 for n=3 circles
Jon Wild, Two inequivalent arrangements of 4 circles with same truth table of intersections.
Jon Wild, Email describing the arrangements of 4 circles with same truth table of intersections (see previous link)

Row sums of A261070.
Apart from first term, row sums of triangles A249752, A252158, A285996, A274776, A274777.
See A275923 and A275924 for the connected arrangements. See also A288554-A288568.
Cf. A241600, A090338, A090339, A048872, A048873, A003036, A132346.
Cf. also A000081, A274702, A274818, A274819, A285995, A287149.
Cf. A132101 (one-dimensional analog).

a(4) is 173, not 168. Corrected by Jon Wild, Aug 08 2015

A duplicate pair of configurations in an older file was spotted by Manfred Scheucher, Aug 13 2016. The pdf and svg files here are now correct.

A241600 Number of ways of arranging n lines in the (affine) plane.

Original entry on oeis.org

1, 1, 2, 4, 9, 47, 791, 37830

Offset: 0

Max Alekseyev and N. J. A. Sloane, May 15 2014

This is in the affine plane, rather than the projective plane, so lines are either parallel or meet in one point.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point. Turning over is also allowed.
a(n) might be called the size of the moduli space of n lines in the affine plane.
The subsequence giving the number of arrangements G_n of n lines in "general position" (with every two lines meeting in one point and every intersection point lying on exactly two lines) is given by A090338.
The moduli space of n points in the affine plane has been studied by several people (see for example Haiman and Miller, 2004; Martin, 2003). There is no direct connection with this problem, but these references are included for background information. - N. J. A. Sloane, Sep 13 2014
Lukas Finschi points out (email, Sep 19 2014) that a(n) = A063859(n)+1 for n <= 7 (but not for larger n). - N. J. A. Sloane, Sep 20 2014

			Let P_n = n parallel lines, S_n = star of n lines through a point, G_n = n lines in general position, L = P_1 = S_1 = G_1 = a single line.
a(1) = 1: L.
a(2) = 2: P_2, S_2.
a(3) = 4: P_3, P_2 L, S_3, G_3.
See link for illustrations of first 5 terms.

B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.

Lukas Finschi, Homepage of Oriented Matroids
L. Finschi and K. Fukuda, Complete combinatorial generation of small point set configurations and hyperplane arrangements, pp. 97-100 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
Stefan Forcey, Planes and axioms, Univ. Akron (2024). See p. 3.
Stefan Forcey, Counting plane arrangements via oriented matroids, arXiv:2504.11461 [math.HO], 2025. See pp. 5, 18.
Komei Fukuda, Hiroyuki Miyata, and Sonoko Moriyama Complete Enumeration of Small Realizable Oriented Matroids, arXiv:1204.0645 [math.CO], 2012; Discrete Comput. Geom. 49 (2013), no. 2, 359--381. MR3017917. (Further background information.)
Mark Haiman, with an Appendix by Ezra Miller, Commutative algebra of n points in the plane. Trends Commut. Algebra, MSRI Publ 51 (2004): 153-180. (Background)
Sergey Kalmykov, Isolated visible infinite straight lines and their combinations, 1920-1922, private collection, on display at Tretyakov gallery. [illustrates a(1)-a(4), and part of a(5)]
J. L. Martin, The slopes determined by n points in the plane. (Background)
Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006; Duke Math. J. 131 (2006), no. 1, 119-165. (Background)
N. J. A. Sloane, Illustration of a(1)-a(5)
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences (2015 talk slides)

Cf. A090338 (lines in general position), A090339 (curved lines in general position), A250001 (circles).

Cf. also A063859, A003036, A048872, A048873, A132346.

a(n) >= A000041(n). - Pablo Hueso Merino, May 10 2021

a(6) and a(7) from Lukas Finschi, Sep 19 2014

A003036 Number of simplicial arrangements of n lines in the plane (the lines do not pass through a common point, all cells are triangles).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 2, 4, 5, 5, 6

Offset: 3

N. J. A. Sloane

J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Stefan Felsner and Jacob E. Goodman, Pseudoline Arrangements, Chapter 5 of Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [Specific reference for this sequence] - _N. J. A. Sloane_, Nov 14 2023
Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors, Handbook of Discrete and Computational Geometry, CRC Press, 2017, see Table 5.6.1. [General reference for 2017 edition of the Handbook] - _N. J. A. Sloane_, Nov 14 2023

Cf. A048872, A048873.

A048873 Number of non-isomorphic arrangements of n lines in the real projective plane such that the lines do not pass through a common point and no point belongs to more than 2 lines.

Original entry on oeis.org

1, 1, 1, 4, 11

Offset: 3

N. J. A. Sloane

These are called simple arrangements.

J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 6.

Cf. A003036, A048872.

The next term is conjectured to be 135.

A132346 Number of nonisomorphic arrangements of n lines in the real projective plane, including the arrangement where all the lines pass through a common point.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 18

Offset: 0

N. J. A. Sloane, Nov 09 2007

See the illustration of initial terms in A048872.

J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.
B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.

Equals A048872 + 1. Cf. A003036, A048873. Different from A121510.

A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

Original entry on oeis.org

0, 0, 1, 2, 5, 7, 11, 15, 21, 25, 32, 38, 47

Offset: 1

N. J. A. Sloane

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]
------------------------------
     0    0
     0    0
     1    1
     2    2
     5    5
     7    7
    11   11
    15   16
    21   21
    25   25  [Grünbaum]
    32   33  [32-triangle solutions found by Honma and Kabanovitch; proved maximal by Savchuk 2025]
    38   38  [Kabanovitch]
    47   47  [Kabanovitch]
 >= 53   54  [Bader]
    65   65  [Suzuki]
    72   72  [Bader]
    85   85  [Bader]
 >= 93   94  [Bader]
   107  107  [Wood]
>= 116  117  [Wood]
   133  133  [Savchuk]
>= 143  144  [Savchuk]
   161  161  [Savchuk]
>= 172  173  [Savchuk]
   191  191  [Bartholdi]
     ?  205
   225  225  [Savchuk]
     ?  239
   261  261  [Bartholdi]
     ?  276
   299  299  [Wood]
     ?  316
   341  341  [Bartholdi]
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.
Named after the Japanese puzzle expert and mathematics teacher Kobon Fujimura (1903-1983). - Amiram Eldar, Jun 19 2021

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

Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", 2008, in Goodman, Jacob E.; Pach, János; Pollack, Richard (eds.), Surveys on Discrete and Computational Geometry: Proceedings of the 3rd AMS-IMS-SIAM Joint Summer Research Conference "Discrete and Computational Geometry—Twenty Years Later" held in Snowbird, UT, June 18-22, 2006, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 105-116, doi:10.1090/conm/453/08797, ISBN 978-0-8218-4239-3, MR 2405679
Martin 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. 1-2, June 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Johannes Bader, Kobon Triangles.
Johannes Bader, Kobon Triangles. [Cached copy, with permission, pdf format]
Johannes Bader, Illustration showing a(17)=85, Nov 28 2007.
Johannes Bader, Illustration showing a(17)=85, Nov 28 2007. [Cached copy, with permission]
Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles, arXiv:0706.0723 [math.CO], 2007.
Nicolas Bartholdi, Jérémy Blanc, Sébastien Loisel, and Pavlo Savchuk, Illustration showing a(33) = 341, 2008.
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]
Martin Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
S. Honma, 三角形の最大数
S. Honma, Illustration showing a(10)>=25
S. Honma, Illustration showing a(11)>=32
S. Honma, n本の直線でn(n-2)/3個の三角形が出来る条件についての考察: part 1, part 2, part 3.
Ed Pegg, Jr., Kobon triangles, 2006.
Ed Pegg, Jr., Kobon Triangles, 2006. [Cached copy, with permission, pdf format]
Luis Felipe Prieto-Martínez, A list of problems in Plane Geometry with simple statement that remain unsolved, arXiv:2104.09324 [math.HO], 2021.
Pavlo Savchuk, Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening, arXiv:2507.07951 [math.CO], 2025. See pp. 1, 17.
Pavlo Savchuk, Illustration showing a(21) = 133
Pavlo Savchuk, Illustration showing a(23) = 161
Pavlo Savchuk, Corresponding pseudo-line arrangement for the optimal n=23 solution
Pavlo Savchuk, Illustration showing a(27) = 225
Pavlo Savchuk, Illustration showing a(22) >= 143
Pavlo Savchuk, Illustration showing a(24) >= 172
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.
Kyle Wood, Illustration showing a(19) = 107
Kyle Wood, Illustration showing a(20) >= 116
Kyle Wood, Illustration showing a(31) = 299

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
The upper bound for even n can be improved: floor(n(n-7/3)/3), proven by Bartholdi et. al.

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".
a(11)-a(13) from Eric W. Weisstein, Jul 26 2025

cp's OEIS Frontend

A250001 Number of arrangements of n circles in the affine plane.

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A241600 Number of ways of arranging n lines in the (affine) plane.

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A003036 Number of simplicial arrangements of n lines in the plane (the lines do not pass through a common point, all cells are triangles).

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A048873 Number of non-isomorphic arrangements of n lines in the real projective plane such that the lines do not pass through a common point and no point belongs to more than 2 lines.

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A132346 Number of nonisomorphic arrangements of n lines in the real projective plane, including the arrangement where all the lines pass through a common point.

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A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

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