A331450 -id:A331450 - 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

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

A331451 Triangle read by rows: Take an 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, ..., n.

Original entry on oeis.org

1, 4, 0, 10, 0, 1, 18, 6, 0, 0, 35, 7, 7, 0, 1, 56, 24, 0, 0, 0, 0, 90, 36, 18, 9, 0, 0, 1, 120, 90, 10, 0, 0, 0, 0, 0, 176, 132, 44, 22, 0, 0, 0, 0, 1, 276, 168, 0, 0, 0, 0, 0, 0, 0, 0, 377, 234, 117, 39, 0, 13, 0, 0, 0, 0, 1, 476, 378, 98, 0, 0, 0, 0, 0, 0, 0, 0, 0, 585, 600, 150, 105, 15, 0, 0, 0, 0, 0, 0, 0, 1, 848, 672, 128, 48, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 3

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

Computed by Scott R. Shannon, Jan 24 2020

			A hexagon with all diagonals drawn contains 18 triangles, 6 quadrilaterals, and no pentagons or hexagons, so row 6 is [18, 6, 0, 0].
Triangle begins:
1,
4,0,
10,0,1,
18,6,0,0,
35,7,7,0,1,
56,24,0,0,0,0,
90,36,18,9,0,0,1,
120,...
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 45
N. J. A. Sloane, Illustration for row n=9. [9-gon with one representative for each type of polygonal cell labeled with its number of sides]

Cf. A007678, A062361.

See A331450 for a version of this triangle in which trailing zeros in the rows have been omitted.

By counting edges in two ways, we have the identity Sum_k k*T(n,k) + n = 2*A135565(n). E.g. for n=7, 3*35+4*7+5*7+6*0+7*1+7 = 182 = 2*A135565(7).

A101363 In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.

Original entry on oeis.org

0, 1, 8, 20, 60, 112, 208, 216, 480, 660, 864, 1196, 1568, 2250, 2464, 2992, 3924, 4332, 5160, 8148, 7040, 8096, 10560, 10600, 12064, 15552, 15288, 17052, 25320, 21080, 23360, 30360, 28288, 30940, 36288, 36852, 40128, 50076, 47120, 50840, 67620

Offset: 2

Graeme McRae, Dec 26 2004, revised Feb 23 2008, Feb 26 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.
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.
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 points 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."

			a(6)=60 because inside a regular 12-gon there are 60 points (4 on each radius and 1 midway between radii) where exactly three diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 2..10000 (terms 2..105 from Graeme McRae)
M. F. Hasler, Interactive illustration of A006561(n)
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
M. Rubinstein, Drawings for n=4,5,6,...
N. J. A. Sloane, Illustrations of a(8) and a(9)
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).]
R. G. Wilson V, Illustration of a(10)
Index entry for Sequences formed by drawing all diagonals in regular polygon

Cf. A006561, A007678, A101364, A101365
A column of A292105.
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. A292104: number of 2-way intersections in the interior of a regular n-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
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon.

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

A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

0, 1, 3, 8, 30, 78, 189, 320, 684, 1010, 1815, 2052, 3978, 4718, 7665, 8576, 13464, 12546, 22059, 23720, 34230, 36542, 50853, 47928, 72900, 76466, 101439, 105560, 137634, 115230, 182745, 188672, 238128, 245378, 305235, 294948, 385614, 395390, 480909, 491840, 592860, 544950, 723303, 737528

Offset: 1

Scott R. Shannon, Jun 02 2021

nonn

See A344857 for other examples and images of the polygons.

			a(3) = 3 as the connected vertices form a triangle with three edges. Six infinite edges between the outer regions are also formed but these are not counted.
a(5) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer regions are also formed.

J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Cf. A344907 (number of edges for odd n), A344857 (number of polygons), A146212 (number of vertices), A344866, A344311, A007678, A331450, A344938.

Bisections: A344907, A347322.

Conjectured formula odd n: a(n) = (n^4 - 7*n^3 + 17*n^2 - 11*n)/4 = (n-1)*n*(n^2-6*n+11)/4.
This formula is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021
See also A344907.
a(n) = A344857(n) + A146212(n) - 1 (Euler's theorem.).

A350000 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of cells in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 11, 1, 1, 4, 21, 24, 1, 1, 4, 31, 42, 50, 1, 1, 4, 41, 42, 190, 80, 1, 1, 4, 51, 42, 400, 152, 154, 1, 1, 4, 61, 42, 680, 152, 802, 220, 1, 1, 4, 71, 42, 1030, 152, 1792, 590, 375, 1, 1, 4, 81, 42, 1450, 152, 2962, 690, 2091, 444, 1

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Dec 06 2021

We use "cell" in the sense of planar graph theory, meaning a "region" or two-dimensional face.
We start at generation 0 with a regular n-gon with a single cell.
At each stage the mitosis process splits each cell into smaller cells by drawing chords between every pair of points on the boundary of that cell.
For the first few generations of mitosis of a triangle, square, pentagon, and hexagon, see the sketch in one of the links below.
The process of going from generation 0 to generation 1 was analyzed by Poonen and Rubinstein (1998) - see A007678 and A331450.
It is worth enlarging the illustrations in order to see the detailed structure and the cell counts in the upper left corner. The illustrations for the mitosis of a 7-gon can be seen in A349808 and are not repeated here.
Conjecture 1: For a fixed value of n, there are integers r and s, which are small compared to n, such that T(n,k) is a polynomial in k of degree r for all k >= s.
For example, T(11,k) = 220*k^2 + 1452*k - 1693 for k >= 2. See the Formulas section below for further examples.
Note that if n is odd, all generations of mitosis of a regular n-gon contain a (smaller) regular n-gon at their center.
Conjecture 2: Apart from the central n-gon when n is odd, any cell will eventually split into a mixture of triangles and pentagons.
If we think of triangles and pentagons are harmless cells, and all other cells as dangerous, the conjecture states that (with the exception of the central odd cells), all cells eventually become harmless.

			The table begins:
.
      |               Number of polygons after k generations
  n\k | 0,    1,     2,     3,      4,      5,      6,      7,      8,      9, ...
----------------------------------------------------------------------------------
   3  | 1,    1,     1,     1,      1,      1,      1,      1,      1,      1, ...
   4  | 1,    4,     4,     4,      4,      4,      4,      4,      4,      4, ...
   5  | 1,   11,    21,    31,     41,     51,     61,     71,     81,     91, ...
   6  | 1,   24,    42,    42,     42,     42,     42,     42,     42,     42, ...
   7  | 1,   50,   190,   400,    680,   1030,   1450,   1940,   2500,   3130, ...
   8  | 1,   80,   152,   152,    152,    152,    152,    152,    152,    152, ...
   9  | 1,  154,   802,  1792,   2962,   4312,   5842,   7552,   9442,  11512, ...
  10  | 1,  220,   590,   690,    790,    890,    990,   1090,   1190,   1290, ...
  11  | 1,  375,  2091,  4643,   7635,  11067,  14939,  19251,  24003,  29195, ...
  12  | 1,  444,   948,   948,    948,    948,    948,    948,    948,    948, ...
  13  | 1,  781,  5461, 14119,  24727,  37285,  51793,  68251,  86659, 107017, ...
  14  | 1,  952,  3066,  4046,   5026,   6006,   6986,   7966,   8946,   9926, ...
  15  | 1, 1456,  9361, 22756,  40186,  61066,  85396, 113176, 144406, 179086, ...
  16  | 1, 1696,  6096,  8240,   9520,  10800,  12080,  13360,  14640,  15920, ...
  17  | 1, 2500, 18225, 49131,  90883, 143175, 206007, 279379, 363291, 457743, ...
  18  | 1, 2466,  7344, 10872,  14166,  16866,  19566,  22266,  24966,  27666, ...
  19  | 1, 4029, 29356, 77616, 140316, 217456, 309036, 415056, 535516, 670416, ...
  20  | 1, 4500, 19580, 31620,  39820,  48020,  56220,  64420,  72620,  80820, ...
  21  | 1, 6175, 40720, 97336, 168022, 252778, 351604, 464500, 591466, 732502, ...
  22  | 1, 6820, 31042, 52030,  65890,  79750,  93610, 107470, 121330, 135190, ...
.

B. Poonen and M. 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.
Scott R. Shannon, Extended Table of A350000 for 5 <= n <= 46, Dec 22, 2021 [This shows the initial terms of the rows in human order (not by antidiagonals)]
Scott R. Shannon, Illustration for T(9,1)
Scott R. Shannon, T(9,2)
Scott R. Shannon, T(9,3)
Scott R. Shannon, T(9,4)
Scott R. Shannon, T(10,1)
Scott R. Shannon, T(10,2)
Scott R. Shannon, T(10,3)
Scott R. Shannon, T(11,1)
Scott R. Shannon, T(11,2)
Scott R. Shannon, T(11,3)
Scott R. Shannon, T(14,1)
Scott R. Shannon, T(14,2)
Scott R. Shannon, T(14,3)
Scott R. Shannon, T(17,3)
Scott R. Shannon, T(29,1)
Scott R. Shannon, Close-up of the 11-gon in T(29,1)
Scott R. Shannon, Close-up of a 9-gon in T(29,2). This shows the mitosis of the 11-gon from generation 1 in the previous image -- it has formed one 9-gon, five 7-gons, twelve 6-gons and numerous other 5, 4, and 3-gons.
Scott R. Shannon, Zoomed-in view of T(51,2). This shows the complicated structure formed after just 2 generations, typical of larger values of n.
N. J. A. Sloane, Rough sketch of first few generations of mitosis of a triangle, square, pentagon, and hexagon. The central pentagonal cell of the pentagon splits into 10 triangles and a pentagon at every generation, with the cells getting smaller and smaller. The third splitting is drawn in red ink. The second splitting of the hexagon is also drawn in red ink, but then all the cells are triangles, and no further mitosis takes place.

Cf. A007678 (column 1), A349807 (column 2), A017281 (row 5), A349808 (row 7); also A350501, A350502.

Cf. also A331450, A349967, A349968.

Formulas for the initial rows: (These are easy to prove.)
To avoid double subscripts, we use a(k) for T(n,k) when we are looking at row n.
n=3: a(k) = 1, for k >= 0.
n=4: a(0) = 1, a(k) = 4 for k >= 1.
n=5: a(k) = 10k+1, k >= 0. See A017281.
n=6: a(0) = 1, a(1) = 24, a(k) = 42 for k >= 2.
n=7: a(0) = 1, a(k) = 35*k^2+35*k-20 for k >= 1. See A349808.
n=8: a(0) = 1, a(1) = 80, a(k) = 152 for k >= 2.
n=9: a(0) = 1, a(1) = 154, a(k) = 90*k^2+540*k-638 for k >= 2.
n=10: a(0) = 1, a(1) = 220, a(k) = 100*k+390 for k >= 2.
n=11: a(0) = 1, a(1) = 375, a(k) = 220*k^2 + 1452*k - 1693 for k >= 2.
n=12: a(0) = 1, a(1) = 444, a(k) = 948 for k >= 2.
n=13: a(0) = 1, a(1) = 781, a(k) = 975*k^2 + 3783*k - 6005 for k >= 2.
n=14: a(0) = 1, a(k) = 980*k + 1106 for k >= 1.
n=15: a(k) = 1725*k^2+5355*k-8834 for k >= 3.
n=16: a(k) = 1280*k + 4400 for k >= 3.
n=18: a(k) = 2700*k + 3366 for k >= 4.
Also T(n,1) = A007678(n).

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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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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A331451 Triangle read by rows: Take an 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, ..., n.

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A101363 In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.

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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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