A101363 -id:A101363 - cp's OEIS frontend

A007678 Number of regions in regular n-gon with all diagonals drawn.

0, 0, 1, 4, 11, 24, 50, 80, 154, 220, 375, 444, 781, 952, 1456, 1696, 2500, 2466, 4029, 4500, 6175, 6820, 9086, 9024, 12926, 13988, 17875, 19180, 24129, 21480, 31900, 33856, 41416, 43792, 52921, 52956, 66675, 69996, 82954, 86800, 102050, 97734, 124271, 129404, 149941

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

This sequence and A006533 are two equivalent ways of presenting the same sequence.
A quasipolynomial of order 2520. - Charles R Greathouse IV, Jan 15 2013
Also the circuit rank of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
This sequence only counts polygons, in contrast to A006533 which also counts the n segments of the circumscribed circle delimited by the edges of the regular n-gon. Therefore a(n) = A006533(n) - n. See also A006561 which counts the intersection points, and A350000 which considers iterated "cutting along diagonals". - M. F. Hasler, Dec 13 2021
The Petrie polygon orthographic projection of a regular n-simplex is a regular (n+1)-gon with all diagonals drawn. Hence a(n+1) is the number of regions in the Petrie polygon of a regular n-simplex. - Mohammed Yaseen, Nov 05 2022

Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
C. A. Pickover, The Mathematics of Oz, Problem 58 "The Beauty of Polygon Slicing", Cambridge University Press, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Robert Dougherty-Bliss, First draft of Python program to produce colored drawings of these figures, Github, Feb 09 2020.
Martin Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
M. F. Hasler, Interactive illustration of A006561(n) & A006533(n); colored version for n=6 and for n=8.
Sascha Kurz, m-gons in regular n-gons (in German).
J. Meeus & N. J. A. Sloane, Correspondence, 1974-1975
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11, no. 1 (1998), pp. 135-156; DOI:10.1137/S0895480195281246. [Author's copy]. The latest arXiv version arXiv:math/9508209 has corrected some typos in the published version.
Bjorn Poonen and Michael Rubinstein, Mathematica programs for these sequences
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
M. Rubinstein, Drawings for n=4,5,6,...
Scott R. Shannon, Colored illustration for n = 17
Scott R. Shannon, Colored illustration for n = 18
Scott R. Shannon, Colored illustration for n = 19
Scott R. Shannon, Colored illustration for n = 23
Scott R. Shannon, Colored illustration for n = 27
Scott R. Shannon, Colored illustration for n = 40
Scott R. Shannon, Colored illustration for n = 41 (1st version)
Scott R. Shannon, Colored illustration for n = 41 (2nd version)
Scott R. Shannon, Colored illustration for n = 41 (3rd version). This variation has coloring based on the number of edges of the polygon: red = 3-gon, orange = 4-gon, yellow = 5-gon, light-green = 6-gon etc.
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18.
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Prasad Balakrishnan Warrier, The physiognomy of the Erdős-Szekeres conjecture (happy ending problem), Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
Eric Weisstein's World of Mathematics, Circuit Rank
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals
Sequences formed by drawing all diagonals in regular polygon
Sequences related to chord diagrams

Cf. A001006, A054726, A006533, A006561, A006600, A007569 (number of vertices), A006522, A135565 (number of line segments).
A062361 gives number of triangles, A331450 and A331451 give distribution of polygons by number of sides.
A333654, A335614, A335646, A337330 give the number of internal n-gon to k-gon contacts for n>=3, k>=n.
A187781 gives number of distinct regions.

del[m_,n_]:=If[Mod[n,m]==0,1,0]; R[n_]:=If[n<3, 0, (n^4-6n^3+23n^2-42n+24)/24 + del[2,n](-5n^3+42n^2-40n-48)/48 - del[4,n](3n/4) + del[6,n](-53n^2+310n)/12 + del[12,n](49n/2) + del[18,n]*32n + del[24,n]*19n - del[30,n]*36n - del[42,n]*50n - del[60,n]*190n - del[84,n]*78n - del[90,n]*48n - del[120,n]*78n - del[210,n]*48n]; Table[R[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

/* Only for odd n > 3, not suitable for other values of n! */ { a(n)=local(nr,x,fn,cn,fn2); nr=0; fn=floor(n/2); cn=ceil(n/2); fn2=(fn-1)^2-1; nr=fn2*n+fn+(n-2)*fn+cn; x=(n-5)/2; if (x>0,nr+=x*(x+1)*(2*x+1)/6*n); nr; } \\ Jon Perry, Jul 08 2003

apply( {A007678(n)=if(n%2, (((n-6)*n+23)*n-42)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 31, 40) +!(n%6)*(310 -53*n))/12 +!(n%12)*49/2 +!(n%18)*32 +!(n%24)*19 -!(n%30)*36 -!(n%42)*50 -!(n%60)*190 -!(n%84)*78 -!(n%90)*48 -!(n%120)*78 -!(n%210)*48)*n)}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A007678(n): return (1176*d(n,12)*n - 3744*d(n,120)*n + 1536*d(n,18)*n - d(n,2)*(5*n**3 - 42*n**2 + 40*n + 48) - 2304*d(n,210)*n + 912*d(n,24)*n - 1728*d(n,30)*n - 36*d(n,4)*n - 2400*d(n,42)*n - 4*d(n,6)*n*(53*n - 310) - 9120*d(n,60)*n - 3744*d(n,84)*n - 2304*d(n,90)*n + 2*n**4 - 12*n**3 + 46*n**2 - 84*n)//48 + 1 # Chai Wah Wu, Mar 08 2021

For odd n > 3, a(n) = sumstep {i=5, n, 2, (i-2)*floor(n/2)+(i-4)*ceiling(n/2)+1} + x*(x+1)*(2*x+1)/6*n), where x = (n-5)/2. Simplifying the floor/ceiling components gives the PARI code below. - Jon Perry, Jul 08 2003
For odd n, a(n) = (24 - 42*n + 23*n^2 - 6*n^3 + n^4)/24. - Graeme McRae, Dec 24 2004
a(n) = A006533(n) - n. - T. D. Noe, Dec 23 2006
For odd n, binomial transform of [1, 10, 29, 36, 16, 0, 0, 0, ...] = [1, 11, 50, 154, ...]. - Gary W. Adamson, Aug 02 2011
a(n) = A135565(n) - A007569(n) + 1. - Max Alekseyev
See the Mma code in A006533 for the explicit Poonen-Rubenstein formula that holds for all n. - N. J. A. Sloane, Jan 23 2020

More terms from Graeme McRae, Dec 26 2004

a(1) = a(2) = 0 prepended by Max Alekseyev, Dec 01 2011

A007569 Number of nodes in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 42, 57, 135, 171, 341, 313, 728, 771, 1380, 1393, 2397, 1855, 3895, 3861, 6006, 5963, 8878, 7321, 12675, 12507, 17577, 17277, 23780, 16831, 31496, 30945, 40953, 40291, 52395, 47017, 66082, 65019, 82290, 80921, 101311, 84883, 123453, 121485

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

I.e., vertex count of the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08 2018
Also the circumference of the n-polygon diagonal intersection graph (since these graphs are Hamiltonian). - Eric W. Weisstein, Mar 08 2018
a(n) = n + sum of row n of triangle A292105. - N. J. A. Sloane, Jun 01 2025

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006; arXiv version, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
M. Rubinstein, Drawings for n=4,5,6,...
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
Eric Weisstein's World of Mathematics, Graph Circumference
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Eric Weisstein's World of Mathematics, Vertex Count
Robert G. Wilson v, Illustration of a(10)
Sequences formed by drawing all diagonals in regular polygon

Cf. A006561, A007678 (regions), A292105.

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

del[m_,n_]:=If[Mod[n,m]==0,1,0]; Int[n_]:=If[n<4, n, n + Binomial[n,4] + del[2,n](-5n^3+45n^2-70n+24)/24 - del[4,n](3n/2) + del[6,n](-45n^2+262n)/6 + del[12,n]*42n + del[18,n]*60n + del[24,n]*35n - del[30,n]*38n - del[42,n]*82n - del[60,n]*330n - del[84,n]*144n - del[90,n]*96n - del[120,n]*144n - del[210,n]*96n]; Table[Int[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

apply( {A007569(n)=A006561(n)+n}, [1..44]) \\ M. F. Hasler, Aug 06 2021

def d(n,m): return not n % m
def A007569(n): return 2 if n == 2 else n*(42*d(n,12) - 144*d(n,120) + 60*d(n,18) - 96*d(n,210) + 35*d(n,24)- 38*d(n,30) - 82*d(n,42) - 330*d(n,60) - 144*d(n,84) - 96*d(n,90)) + (n**4 - 6*n**3 + 11*n**2 + 18*n -d(n,2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n,4)*n - 4*d(n,6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021

a(n) = A006561(n)+n. - T. D. Noe, Dec 23 2006

If n is odd, a(n) = binomial(n,4) + n. - N. J. A. Sloane, Aug 30 2021

A006561 Number of intersections of diagonals in the interior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 1, 5, 13, 35, 49, 126, 161, 330, 301, 715, 757, 1365, 1377, 2380, 1837, 3876, 3841, 5985, 5941, 8855, 7297, 12650, 12481, 17550, 17249, 23751, 16801, 31465, 30913, 40920, 40257, 52360, 46981, 66045, 64981, 82251, 80881, 101270, 84841, 123410, 121441

Offset: 1

N. J. A. Sloane, Bjorn Poonen (poonen(AT)math.princeton.edu)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
Johan Gielis and Ilia Tavkhelidze, The general case of cutting of GML surfaces and bodies, arXiv:1904.01414 [math.GM], 2019.
Jessica Gonzalez, Illustration of a(4) through a(9).
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.
M. F. Hasler, Interactive illustration of A006561(n), Sep 01 2017. (For colored versions see A006533.)
Sascha Kurz, m-gons in regular n-gons.
Roger Mansuy, Des croisements pas si faciles à compter, La Recherche, 547, Mai 2019 (in French).
B. Poonen and M. Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, No.1 (1998) pp. 135-156; DOI:10.1137/S0895480195281246. [Copy on B. Poonen's web site.]
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]: revision from 2006 has a few typos from the published version corrected.
B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences.
M. Rubinstein, Drawings for n=4,5,6,....
N. J. A. Sloane, Illustrations of a(8) and a(9).
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 18.
Robert G. Wilson v, Illustration of a(10)
Index entry for Sequences formed by drawing all diagonals in regular polygon

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
See also A101363, A292104, A292105.
See A290447 for an analogous problem on a line.

delta:=(m,n) -> if (n mod m) = 0 then 1 else 0; fi;
f:=proc(n) global delta;
if n <= 2 then 0 else \
binomial(n,4)  \
+ (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24 \
- (3*n/2)*delta(4,n) \
+ (-45*n^2 + 262*n)*delta(6,n)/6  \
+ 42*n*delta(12,n) \
+ 60*n*delta(18,n) \
+ 35*n*delta(24,n) \
- 38*n*delta(30,n) \
- 82*n*delta(42,n) \
- 330*n*delta(60,n) \
- 144*n*delta(84,n) \
- 96*n*delta(90,n) \
- 144*n*delta(120,n) \
- 96*n*delta(210,n); fi; end;
[seq(f(n),n=1..100)]; # N. J. A. Sloane, Aug 09 2017

del[m_,n_]:=If[Mod[n,m]==0,1,0]; Int[n_]:=If[n<4, 0, Binomial[n,4] + del[2,n](-5n^3+45n^2-70n+24)/24 - del[4,n](3n/2) + del[6,n](-45n^2+262n)/6 + del[12,n]*42n + del[18,n]*60n + del[24,n]*35n - del[30,n]*38n - del[42,n]*82n - del[60,n]*330n - del[84,n]*144n - del[90,n]*96n - del[120,n]*144n - del[210,n]*96n]; Table[Int[n], {n,1,1000}] (* T. D. Noe, Dec 21 2006 *)

apply( {A006561(n)=binomial(n,4)+if(n%2==0, (n>2) + (-5*n^2+45*n-70)*n/24 + vecsum([t[2] | t<-[4,6,12,18,24,30,42,60,84,90,120,210;-3/2,(262-45*n)/6,42,60,35,-38,-82,-330,-144,-96,-144,-96], n%t[1]==0])*n)}, [1..44]) \\ M. F. Hasler, Aug 23 2017, edited Aug 06 2021

def d(n,m): return not n % m
def A006561(n): return 0 if n == 2 else n*(42*d(n,12) - 144*d(n,120) + 60*d(n,18) - 96*d(n,210) + 35*d(n,24)- 38*d(n,30) - 82*d(n,42) - 330*d(n,60) - 144*d(n,84) - 96*d(n,90)) + (n**4 - 6*n**3 + 11*n**2 - 6*n -d(n,2)*(5*n**3 - 45*n**2 + 70*n - 24) - 36*d(n,4)*n - 4*d(n,6)*n*(45*n - 262))//24 # Chai Wah Wu, Mar 08 2021

Let delta(m,n) = 1 if m divides n, otherwise 0.
For n >= 3, a(n) = binomial(n,4) + (-5*n^3 + 45*n^2 - 70*n + 24)*delta(2,n)/24
- (3*n/2)*delta(4,n) + (-45*n^2 + 262*n)*delta(6,n)/6 + 42*n*delta(12,n)
+ 60*n*delta(18,n) + 35*n*delta(24,n) - 38*n*delta(30,n)
- 82*n*delta(42,n) - 330*n*delta(60,n) - 144*n*delta(84,n)
- 96*n*delta(90,n) - 144*n*delta(120,n) - 96*n*delta(210,n). [Poonen and Rubinstein, Theorem 1] - N. J. A. Sloane, Aug 09 2017
For odd n, a(n) = binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24, see A053126. For even n, use this formula, but then subtract 2 for every 3-crossing, subtract 5 for every 4-crossing, subtract 9 for every 5-crossing, etc. The number to be subtracted for a d-crossing is (d-1)*(d-2)/2. - Graeme McRae, Dec 26 2004
a(n) = A007569(n) - n. - T. D. Noe, Dec 23 2006
a(2n+5) = A053126(n+4). - Philippe Deléham, Jun 07 2013

A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 4, 10, 0, 1, 18, 6, 35, 7, 7, 0, 1, 56, 24, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 176, 132, 44, 22, 276, 168, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1404, 954, 72, 18, 18, 1653, 1444, 646, 190, 57, 38, 2200, 1580, 580, 120, 0, 20, 2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 25 2020

Computed by Scott R. Shannon, Jan 24 2020
See A331451 for a version of this triangle giving the counts for k = 3 through n.
Mitosis of convex polygons, by Scott R. Shannon and N. J. A. Sloane, Dec 13 2021 (Start)
Borrowing a term from biology, we can think of this process as the "mitosis" of a regular polygon. Row 6 of this triangle shows that a regular hexagon "mitoses" into 18 triangles and 4 quadrilaterals, which we denote by 3^18 4^6.
What if we start with a convex but not necessarily regular n-gon? Let M(n) denote the number of different decompositions into cells that can be obtained. For n = 3, 4, and 5 there is only one possibility. For n = 6 there are two possibilities, 3^18 4^6 and 3^19 4^3 5^3. For n = 7 there are at least 11 possibilities. So the sequence M(n) for n >= 3 begins 1, 1, 1, 2, >=11, ...
The links below give further information. See also A350000. (End)

			A hexagon with all diagonals drawn contains 18 triangles and 6 quadrilaterals, so row 6 is [18, 6].
Triangle begins:
  1,
  4,
  10, 0, 1,
  18, 6,
  35, 7, 7, 0, 1,
  56, 24,
  90, 36, 18, 9, 0, 0, 1,
  120, 90, 10,
  176, 132, 44, 22, 0, 0, 0, 0, 1
  276, 168,
  377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1,
  476, 378, 98,
  585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1,
  848, 672, 128, 48,
  1054, 901, 357, 136, 17, 34, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  1404, 954, 72, 18, 18,
  1653, 1444, 646, 190, 57, 38, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
  2200, 1580, 580, 120, 0, 20,
  2268, 2520, 903, 462, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  2992, 2860, 814, 66, 44, 44,
  3749, 2990, 1564, 644, 115, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
  ...
The row sums are A007678, the first column is A062361.

M. Rubinstein, Drawings of A007678 for n=4,5,6,....
Scott R. Shannon, Rows 3 through 596 of A331450.
N. J. A. Sloane, Illustration for row n=9 of A331450. [9-gon with one representative for each type of polygonal cell labeled with its number of sides].
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Scott R. Shannon and N. J. A. Sloane, N. J. A. Sloane, Table showing number of ways to "mitose" a convex n-gon. [From here on the links are related to the mitosis problem, and are in logical rather than alphabetical order]
Scott R. Shannon, Illustration for mitosis 5.1 of a pentagon (the cell counts are the same whether of not the pentagon is regular)
Scott R. Shannon, Illustration for mitosis 6.1 of a hexagon with a triple point (the cell counts are the same whether of not the hexagon is regular, as long as it has a triple point)
Scott R. Shannon, Illustration for mitosis 6.2 of a hexagon (without a triple point)
N. J. A. Sloane, Illustration for mitosis 7a of a 7-gon
Scott R. Shannon, A second (colored) illustration for mitosis 7a of a 7-gon
Scott R. Shannon, Illustration for mitosis 7b
Scott R. Shannon, Illustration for mitosis 7c
Scott R. Shannon, Illustration for mitosis 7d
Scott R. Shannon, Illustration for mitosis 7e
Scott R. Shannon, Illustration for mitosis 7f
Scott R. Shannon, Illustration for mitosis 7g
Scott R. Shannon, Illustration 1 for mitosis 7h
Scott R. Shannon, Illustration 2 for mitosis 7h (same cell counts as preceding illustration but a different polygon - look at the brown cells)
Scott R. Shannon, Illustration for mitosis 7i
Scott R. Shannon, Illustration for mitosis 7j
Scott R. Shannon, Illustration for mitosis 7k
M. F. Hasler, Nine examples of dissections of convex 7-gons [These are all subsumed in the above illustrations]

Cf. A007678, A062361, A331451, A341729, A341730, A335000.

Added "regular" to definition. - N. J. A. Sloane, Mar 06 2021

A062361 Number of triangular regions in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 4, 10, 18, 35, 56, 90, 120, 176, 276, 377, 476, 585, 848, 1054, 1404, 1653, 2200, 2268, 2992, 3749, 4416, 5000, 6292, 6777, 8316, 9222, 11670, 11501, 14368, 15840, 18598, 19705, 24444, 25012, 28842, 30966, 36000, 39278, 45318, 46999, 53900

Offset: 3

Sascha Kurz, Jul 07 2001

Also the number of 3-cycles and maximum cliques in the n-polygon diagonal intersection graph. - Eric W. Weisstein, Mar 08-09 2018

			a(4) = 4 because in a quadrilateral the diagonals cross to make four triangles.

Andrew Howroyd, Table of n, a(n) for n = 3..100
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998).
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
S. E. Sommars and T. Sommars, Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, J. Integer Sequences, 1 (1998), #98.1.5.
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Maximum Clique
Eric Weisstein's World of Mathematics, Polygon Diagonal Intersection Graph
Sequences formed by drawing all diagonals in regular polygon

Cf. A006600, A007678.

Cf. A300552 (4-cycles), A300553 (5-cycles), A300554 (6-cycles).

a(n) = n * A067162(n).

A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

Original entry on oeis.org

0, 0, 6, 7, 24, 36, 90, 132, 168, 234, 378, 600, 672, 901, 954, 1444, 1580, 2520, 2860, 2990, 3696, 4800, 5070, 6750, 7644, 9309, 7920, 12927, 12896, 15576, 16898, 20475, 18684, 25382, 27246, 30966, 32760, 37064, 37170, 45838, 47300, 55350, 60996, 69231, 66864, 80507, 87550, 98124, 103272

Offset: 4

Sascha Kurz, Jan 06 2002

nonn

			a(6)=6 because the 6 regions around the center are quadrilaterals.

B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156.

Scott R. Shannon, Table of n, a(n) for n = 4..765
Sascha Kurz, m-gons in regular n-gons
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Mathematica programs for these sequences
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
Sequences formed by drawing all diagonals in regular polygon

Cf. A007678, A067163, A064869, A067152, A067153, A067154, A067155, A067156, A067157, A067158, A067159.

Conjecture: a(n) ~ c * n^4. Is c = 1/64 ? - Bill McEachen, Mar 03 2024

Title clarified, a(47) and above by Scott R. Shannon, Dec 04 2021

A292104 Number of interior points that are the intersections of exactly two chords in the configuration A006561(n).

Original entry on oeis.org

0, 0, 0, 1, 5, 12, 35, 40, 126, 140, 330, 228, 715, 644, 1365, 1168, 2380, 1512, 3876, 3360, 5985, 5280, 8855, 6144, 12650, 11284, 17550, 15680, 23751, 13800, 31465, 28448, 40920, 37264, 52360, 42444, 66045, 60648, 82251, 75720, 101270, 75012, 123410, 114400, 148995, 138644, 178365, 152064

Offset: 1

N. J. A. Sloane, Sep 14 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..120 from Scott R. Shannon)
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]

Cf. A006561. Column k=2 of A292105.

a(31)-a(48) from Scott R. Shannon, Mar 04 2022

A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 5, 0, 12, 1, 0, 35, 0, 40, 8, 1, 0, 126, 0, 140, 20, 0, 1, 0, 330, 0, 228, 60, 12, 0, 1, 0, 715, 0, 644, 112, 0, 0, 0, 1, 0, 1365, 0, 1168, 208, 0, 0, 0, 0, 1, 0, 2380, 0, 1512, 216, 54, 54, 0, 0, 0, 1, 0, 3876, 0, 3360, 480, 0, 0, 0, 0, 0, 0, 1, 0, 5985

Offset: 1

N. J. A. Sloane, Sep 14 2017

			Triangle begins:
  0;
  0;
  0;
  0,   1;
  0,   5;
  0,  12,  1;
  0,  35;
  0,  40,  8,  1;
  0, 126;
  0, 140, 20,  0, 1;
  0, 330;
  0, 228, 60, 12, 0, 1;
See the attached text file for the first 100 rows.

Seiichi Manyama, Rows n = 1..250, flattened
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). [Copy on B. Poonen's web site]
B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)
Scott R. Shannon, Table for n=1..100.
Scott R. Shannon, Image of 8-gon.
Scott R. Shannon, Image of 9-gon.
Scott R. Shannon, Image of 12-gon.

Columns give A292104, A101363 (2n-gon), A101364, A101365.
Row sums give A006561.
Cf. A335102.

a(27) and beyond by Scott R. Shannon, May 15 2022

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.

cp's OEIS Frontend

A007678 Number of regions in regular n-gon with all diagonals drawn.

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A007569 Number of nodes in regular n-gon with all diagonals drawn.

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A006561 Number of intersections of diagonals in the interior of a regular n-gon.

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A331450 Irregular triangle read by rows: Take a regular n-sided polygon (n>=3) with all diagonals drawn, as in A007678. Then T(n,k) = number of k-sided polygons in that figure for k = 3, 4, ..., max_k.

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A062361 Number of triangular regions in regular n-gon with all diagonals drawn.

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A067151 Number of regions in regular n-gon which are quadrilaterals (4-gons) when all its diagonals are drawn.

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A292104 Number of interior points that are the intersections of exactly two chords in the configuration A006561(n).

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A292105 Irregular triangle read by rows: T(n,k) = the number of interior points that are the intersections of exactly k chords in the configuration A006561(n) (n >= 1, k >= 1).

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