A006561 -id:A006561 - cp's OEIS frontend

A053126 Binomial coefficients binomial(2*n-3,4).

5, 35, 126, 330, 715, 1365, 2380, 3876, 5985, 8855, 12650, 17550, 23751, 31465, 40920, 52360, 66045, 82251, 101270, 123410, 148995, 178365, 211876, 249900, 292825, 341055, 395010, 455126, 521855, 595665, 677040, 766480

Offset: 4

Wolfdieter Lang

Number of intersections of diagonals in the interior of regular (2n-3)-gon. - Philippe Deléham, Jun 07 2013

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

Vincenzo Librandi, Table of n, a(n) for n = 4..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Milan Janjic, Two Enumerative Functions University of Banja Luka (Bosnia and Herzegovina, 2017).
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A053123, A002492.

[Binomial(2*n-3,4): n in [4..40]]; // Vincenzo Librandi, Oct 07 2011

Table[Binomial[2*n-3,4], {n,4,50}] (* G. C. Greubel, Aug 26 2018 *)

for(n=4,50, print1(binomial(2*n-3,4), ", ")) \\ G. C. Greubel, Aug 26 2018

a(n) = binomial(2*n-3, 4) if n >= 4 else 0;
G.f.: (5+10*x+x^2)/(1-x)^5.
a(n) = A053123(n,4), n >= 4; a(n) = 0, n=0..3 (fifth column of shifted Chebyshev's S-triangle, decreasing order).
a(n) = A006561(2n-3). - Philippe Deléham, Jun 07 2013
E.g.f.: (90 - 84*x + 39*x^2 - 12*x^3 + 4*x^4)*exp(x)/6. - G. C. Greubel, Aug 26 2018
From Amiram Eldar, Jan 04 2022: (Start)
Sum_{n>=4} 1/a(n) = 34/3 - 16*log(2).
Sum_{n>=4} (-1)^n/a(n) = 2*Pi - 4*log(2) - 10/3. (End)

A101364 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly four diagonals intersect.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 12, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 264, 0, 0, 0, 0, 0, 420, 0, 0, 0, 0, 0, 396, 0, 0, 0, 0, 0, 1134, 0, 0, 0, 0, 0, 1200, 0, 0, 0, 0, 0, 1296, 0, 0, 0, 0, 0, 3780, 0, 0, 0, 0, 0, 2310, 0, 0, 0, 0, 0, 2520, 0, 0, 0, 0, 0, 3276, 0, 0, 0, 0, 0, 3612, 0, 0, 0, 0, 0, 4050

Offset: 3

Graeme McRae, Dec 26 2004, revised Feb 23 2008

nonn

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

			a(18)=54 because inside a regular 18-gon there are 54 points where exactly four diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 3..10000 (terms 3..210 from Graeme McRae)
Sequences formed by drawing all diagonals in regular polygon

A column of A292105.
Cf. A006561, A007678.
Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon.
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.

A101365 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five diagonals intersect.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 216, 0, 0, 0, 0, 0, 546, 0, 0, 0, 0, 0, 336, 0, 0, 0, 0, 0, 648, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 990, 0, 0, 0, 0, 0, 936, 0, 0, 0, 0, 0, 1404, 0, 0, 0, 0, 0, 2352, 0, 0, 0, 0, 0, 1890, 0, 0, 0, 0

Offset: 3

Graeme McRae, Dec 26 2004, revised Feb 23 2008

nonn

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

			a(18)=54 because inside a regular 18-gon there are 54 points (3 on each radius) where exactly five diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 3..10000 (terms 3..210 from Graeme McRae)
Sequences formed by drawing all diagonals in regular polygon

A column of A292105.
Cf. A006561, A007678.
Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.
Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon.
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.

A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 1, 35, 40, 8, 1, 126, 140, 20, 0, 1, 330, 228, 60, 12, 0, 1, 715, 644, 112, 0, 0, 0, 1, 1365, 1168, 208, 0, 0, 0, 0, 1, 2380, 1512, 216, 54, 54, 0, 0, 0, 1, 3876, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 5985, 5280, 660, 0, 0, 0, 0, 0, 0, 0, 1, 8855, 6144, 864, 264, 24, 0, 0, 0, 0, 0, 0, 12, 12650

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 23 2020

			Table begins:
      0;
      0;
      0;
      1;
      5;
     12,    1;
     35;
     40,    8,   1;
    126;
    140,   20,   0,   1;
    330;
    228,   60,  12,   0,   1;
    715;
    644,  112,   0,   0,   0,  1;
   1365;
   1168,  208,   0,   0,   0,  0, 1;
   2380;
   1512,  216,  54,  54,   0,  0, 0, 1;
   3876;
   3360,  480,   0,   0,   0,  0, 0, 0, 1;
   5985;
   5280,  660,   0,   0,   0,  0, 0, 0, 0, 1;
   8855;
   6144,  864, 264,  24,   0,  0, 0, 0, 0, 0, 1;
  12650;
  11284, 1196,   0,   0,   0,  0, 0, 0, 0, 0, 0, 1;
  17550;
  15680, 1568,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 1;
  23751;
  13800, 2250, 420, 180, 120, 30, 0, 0, 0, 0, 0, 0, 0, 1;
  31465;
  28448, 2464,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  40920;
  37264, 2992,   0,   0,   0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  52360;

B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Scott R. Shannon, Image of the vertices for n=5.
Scott R. Shannon, Image of the vertices for n=6.
Scott R. Shannon, Image of the vertices for n=8.
Scott R. Shannon, Image of the vertices for n=12.
Scott R. Shannon, Image of the vertices for n=13.
Scott R. Shannon, Image of the vertices for n=16.
Scott R. Shannon, Image of the vertices for n=18.
Scott R. Shannon, Image of the vertices for n=20.
Scott R. Shannon, Image of the vertices for n=24.
Scott R. Shannon, Image of the vertices for n=30.
Index entries for sequences related to stained glass windows

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A007569, A007678, A053126, A292105, A333275.

If n = 2t+1 is odd then the n-th row has a single term, T(2t+1, 2t+4) = binomial(2t+1,4) (these values are given in A053126).

A137938 Number of 4-way intersections in the interior of a regular 6n-gon.

Original entry on oeis.org

0, 12, 54, 264, 420, 396, 1134, 1200, 1296, 3780, 2310, 2520, 3276, 3612, 4050, 5088, 5712, 5724, 7182, 11400, 9072, 9372, 10626, 11088, 12600, 13260, 14094, 15960, 17052, 23220, 19530, 20928, 21384, 23052, 26250, 25704, 27972, 28956, 30186, 39600, 34440, 34524

Offset: 1

Graeme McRae, Feb 23 2008

nonn

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

			a(3)=54 because there are 54 points in the interior of an 18-gon at which exactly four diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Sequences formed by drawing all diagonals in regular polygon

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.
Cf. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon

a(n) = A101364(6*n). - Seiichi Manyama, Jul 20 2024

More terms from Seiichi Manyama, Jul 20 2024

A137939 Number of 5-way intersections in the interior of a regular 6n-gon.

Original entry on oeis.org

0, 0, 54, 24, 180, 216, 546, 336, 648, 720, 990, 936, 1404, 2352, 1890, 1824, 2448, 2592, 3078, 3720, 4284, 3960, 4554, 4464, 5400, 5616, 6318, 7896, 7308, 7560, 8370, 8256, 9504, 9792, 11550, 10584, 11988, 12312, 13338, 14640, 14760, 17640, 16254, 16104, 17820, 18216, 19458, 19296, 22344, 21600

Offset: 1

Graeme McRae, Feb 23 2008

nonn

When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."

			a(3) = 54 because there are 54 points in the interior of an 18-gon at which exactly five diagonals meet.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Sequences formed by drawing all diagonals in regular polygon

Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon..
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon.
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon.
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon.
Cf. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon.

a(n) = A101365(6*n). - Seiichi Manyama, Jul 20 2024

More terms from Seiichi Manyama, Jul 20 2024

A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

Original entry on oeis.org

0, 1, 12, 30, 128, 147, 264, 1056, 600, 825, 2380, 1482, 1932, 9635, 3024, 3672, 8484, 5301, 6300, 19474, 8580, 9867, 20744, 12900, 14664, 30141, 18564, 20706, 62200, 25575, 28320, 54956, 34272, 37485, 62868, 44622, 48564, 86359, 57000, 61500, 117068, 71337

Offset: 2

Jonathan Sondow, Jul 25 2015

nonn

Same as (total number of triangles visible in convex 2n-gon with all diagonals drawn in general position) - (total number of triangles visible in regular 2n-gon with all diagonals drawn).
Number of triple-crossings of diagonals in the regular 2n+1-gon is 0.
See Sillke 1998 (where a(n) is called "T(2n)") for explanations and extensive annotated references.
See A005732 and A006600 for more comments, references, links, formulas, examples, programs, and lists from which to compute a(n) = A005732(2n) - A006600(2n) up to n = 500.

			With only 2 diagonals in a 4-gon, there can be no triple-crossings, so a(2) = 0.

T. Sillke, Number of triangles for a convex n-gon, 1998.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

a(n) = A005732(2n) - A006600(2n).

A330913 The number of vertices formed by straight line segments mutually connecting all vertices of a semicircular polygon defined in A333642.

Original entry on oeis.org

4, 8, 16, 34, 63, 113, 185, 253, 438, 638, 854, 1228, 1641, 1825, 2783, 3543, 4304, 5508, 6748, 7745, 9859, 11773, 13653, 16409, 19178, 21838, 25770, 29648, 32696, 38683, 43899, 48903, 55916, 62784, 69604, 78378, 87175, 95699, 106993, 118093, 128431, 142838

Offset: 1

Scott R. Shannon and N. J. A. Sloane, May 02 2020

nonn

See A333642 for a precise definition of the polygon and images.

Lars Blomberg, Table of n, a(n) for n = 1..100

Cf. A333642 (regions), A330911 (edges), A330914 (n-gons), A331453, A333026, A006561.

a(21) and beyond from Lars Blomberg, May 03 2020

A364830 a(n) is the number of line intersections (not coinciding with a vertex) inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 9, 0, 1, 4, 0, 0, 8, 0, 11, 4, 1, 0, 29, 0, 1, 0, 11, 0, 43, 0, 0, 4, 1, 8, 50, 0, 1, 4, 39, 0, 49, 0, 11, 24, 1, 0, 73, 0, 12, 4, 11, 0, 33, 8, 43, 4, 1, 0, 193, 0, 1, 28, 0, 8, 57, 0, 11, 4, 71, 0, 164, 0, 1, 28, 11, 12, 61

Offset: 1

Paolo Xausa, Aug 09 2023

nonn

Inspired by an X (or Twitter) post by Matt Henderson (see links section).

Matt Henderson, X post, 2023.
Paolo Xausa, Illustration of first terms.
Paolo Xausa, Illustration of a(60).

Cf. A006561, A007569, A032741, A364828, A364829, A364838.

a(n) = 0 if n is prime, a power of 2 or A032741(n) <= 2.

a(n) = A364838(n) - n.

A364838 a(n) is the number of line intersections (including vertices) inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 21, 13, 15, 19, 16, 17, 26, 19, 31, 25, 23, 23, 53, 25, 27, 27, 39, 29, 73, 31, 32, 37, 35, 43, 86, 37, 39, 43, 79, 41, 91, 43, 55, 69, 47, 47, 121, 49, 62, 55, 63, 53, 87, 63, 99, 61, 59, 59, 253, 61, 63, 91, 64, 73, 123, 67

Offset: 1

Paolo Xausa, Aug 09 2023

nonn

Inspired by an X (or Twitter) post by Matt Henderson (see links section).

Matt Henderson, X post, 2023.
Paolo Xausa, Illustration of first terms.
Paolo Xausa, Illustration of a(60).

Cf. A006561, A007569, A032741, A364828, A364829, A364830.

a(n) = n if n is prime, a power of 2 or A032741(n) <= 2.

a(n) = A364830(n) + n.

cp's OEIS Frontend

A053126 Binomial coefficients binomial(2*n-3,4).

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A101364 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly four diagonals intersect.

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A101365 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five diagonals intersect.

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A335102 Irregular triangle read by rows: consider the regular n-gon defined in A007678. T(n,k) (n >= 1, k >= 4+2*t where t>=0) is the number of non-boundary vertices in the n-gon at which k polygons meet.

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A137938 Number of 4-way intersections in the interior of a regular 6n-gon.

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A137939 Number of 5-way intersections in the interior of a regular 6n-gon.

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A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

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A330913 The number of vertices formed by straight line segments mutually connecting all vertices of a semicircular polygon defined in A333642.

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Extensions

A364830 a(n) is the number of line intersections (not coinciding with a vertex) inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

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