A135565 -id:A135565 - 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

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).

A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

1, 5, 37, 257, 817, 2757, 4825, 12293, 19241, 33549, 49577, 87685, 101981, 178465, 220113, 286357, 379097, 551669, 606241, 880293, 951445, 1209049, 1507521, 1947877, 2002669, 2624409, 3056093, 3550425, 3955049, 5069037, 5062153, 6669665, 7081969, 8143193, 9365089, 10296469

Offset: 0

N. J. A. Sloane, Jan 23 2020

nonn

Equivalently, this is A334690(n) + 4*n.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 20.
Scott R. Shannon, Image for n = 1.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 10.

Cf. A255011, A331448.

For the circular analog see A006533, A007678, A007569, A135565.

By Euler's formula, a(n) = A331448(n) - A255011(n) + 1.

a(11)-a(35) from Giovanni Resta, Jan 28 2020

A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

2, 12, 16, 120, 138, 532, 432, 1548, 1450, 3696, 3120, 7488, 6874, 13560, 12000, 23256, 20970, 36328, 32400, 54852, 51194, 79672, 70752, 113100, 105274, 154764, 144480, 206712, 195810, 272800, 255808, 352176, 335002, 446740, 422784, 560920, 534698, 695760, 659440, 850340, 815682

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
See A358746 and A358782 for images of the circles.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Cf. A358746 (vertices), A358782 (regions), A359009 (k-gons), A135565, A344899.

See allso A370976-A370979.

a(n) = A358746(n) + A358782(n) - 1 by Euler's formula.

A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

Original entry on oeis.org

3, 6, 5, 9, 7, 13, 9, 29, 11, 40, 13, 43, 15, 212, 17, 231, 19

Offset: 3

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

Theorem: If n is odd then a(n) = n.
Proof. (i) If n is odd then the central cell in a regular n-gon with all diagonals drawn is a smaller regular n-gon. So if n is odd, then a(n) <= n.
(ii) Suppose a convex m-gon, not necessarily regular, with all diagonals drawn has a cell with e edges. Each edge when extended meets two vertices, so at most 2e vertices are involved in defining the boundary of that cell.
On the other hand no vertex can define more than two edges of the cell, so 2e <= 2m, so e <= m. So to get an n-sided cell, we need at least n vertices. So a(n) >= n. QED.
If a(20) > 0 it is greater than 765 - Scott R. Shannon, Nov 30 2021

			Examining the images in A007678, for example Michael Rubinstein's illustration, or the images shown here, we see that the first occurrence of a five-sided cell is for m = 5, so a(5) = 5. The first time we see a four-sided cell is for m = 6, so a(4) = 6.

Martin Balko, Anna Brötzner, Fabian Klute, and Josef Tkadlec, Faces in Rectilinear Drawings of Complete Graphs, 40th European Workshop on Computational Geometry, Ioannina, Greece, March 13-15, 2024. See pp. 4, 7.
Scott R. Shannon, Image for a(3) = 3.
Scott R. Shannon, Image for a(4) = 6.
Scott R. Shannon, Image for a(5) = 5.
Scott R. Shannon, Image for a(6) = a(9) = 9.
Scott R. Shannon, Image for a(7) = 7.
Scott R. Shannon, Image for a(8) = a(13) = 13.
Scott R. Shannon, Image for a(10) = 29.
Scott R. Shannon, Image for a(11) = 11.
Scott R. Shannon, Image for a(12) = 40.
Scott R. Shannon, Image for a(14) = 43.
Scott R. Shannon, Image for a(15) = 15.

Cf. A007678, A331450, A331451, A007569, A135565.

See also A341729 and A341730 for the maximum number of sides in any cell.

a(16)-a(19) added by Scott R. Shannon, Mar 14 2021

A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

Original entry on oeis.org

0, 8, 92, 596, 1936, 6020, 11088, 26260, 42144, 72296, 107832, 183340, 222940, 371100, 466936, 609916, 804504, 1139632, 1288536, 1813288, 2012676, 2536572, 3142008, 3997580, 4230340, 5430444, 6331892, 7360512, 8262568, 10367804

Offset: 0

N. J. A. Sloane, Jan 23 2020, based on a comment from Michael De Vlieger in A255011

nonn

By Euler's formula, a(n) = A331449(n) + A255011(n) - 1.

Zhao Hui Du, Table of n, a(n) for n = 0..136 (terms 0..52 from Lars Blomberg).

Cf. A255011, A331449.

For the circular analog see A006533, A007678, A007569, A135565.

a(11)-a(29) from Giovanni Resta, Jan 28 2020

A371375 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of (curved) edges formed.

Original entry on oeis.org

1, 2, 12, 12, 75, 66, 350, 360, 1071, 1150, 2684, 2148, 5603, 5950, 10110, 10928, 18309, 16830, 29564, 30500, 44961, 46882, 66746, 64872, 95125, 97786, 131112, 135156, 177567, 169770, 235042, 240928, 304359, 312086, 389340, 388764, 491175, 503158, 610662, 624280, 752145, 749742, 917276

Offset: 1

Scott R. Shannon, Mar 20 2024

nonn

See A371373 and A371374 for images of the graphs.

Cf. A371373 (vertices), A371374 (regions), A371376 (k-gons), A371377 (vertex crossings), A371255, A135565, A358783, A359047.

a(n) = A371373(n) + A371374(n) - 1 by Euler's formula.

A359047 Number of distinct edges among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

Original entry on oeis.org

1, 4, 12, 84, 120, 330, 504, 1240, 1332, 2850, 3696, 5172, 7176, 10906, 12660, 19280, 22440, 28494, 35796, 46220, 52752, 68662, 79488, 91272, 111000, 136838, 149472, 181972, 204972, 229650, 268212, 317024, 343860, 404090, 441420, 496764, 553224, 636538, 679224, 776200, 839844, 914634, 1017036

Offset: 1

Scott R. Shannon, Dec 14 2022

nonn

See A331702 and A359046 for further details and images.

No formula for a(n) is currently known.

Cf. A331702 (vertices), A359046 (regions), A359061 (k-gons), A358783, A135565.

a(n) = A331702(n) + A359046(n) - 1 by Euler's formula.

A333654 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 4, 10, 0, 5, 18, 12, 28, 14, 21, 0, 7, 56, 48, 54, 54, 54, 36, 0, 0, 9, 180, 160, 30, 88, 154, 88, 88, 0, 0, 0, 0, 11, 240, 336, 130, 260, 299, 182, 0, 104, 0, 0, 0, 0, 13, 266, 616, 266, 180, 600, 375, 300, 90, 0, 0, 0, 0, 0, 0, 0, 15, 448, 1056, 320, 256

Offset: 3

Scott R. Shannon, Aug 23 2020

See A335614 for the number of 4-gon to k-gon contacts, with k>=4.
See A335646 for the number of 5-gon to k-gon contacts, with k>=5.
See A337330 for the number of 6-gon to k-gon contacts, with k>=6.
See A007678 for the number of regions and images of other n-gons.

			The table begins:
.
0;
4;
10,0,5;
18,12;
28,14,21,0,7;
56,48;
54,54,54,36,0,0,9;
180,160,30;
88,154,88,88,0,0,0,0,11;
240,336;
130,260,299,182,0,104,0,0,0,0,13;
266,616,266;
180,600,375,300,90,0,0,0,0,0,0,0,15;
448,1056,320,256;
238,816,935,578,85,238,0,0,0,0,0,0,0,0,17;
900,1836,324,108,126;
304,1520,1463,798,304,228,0,0,0,0,0,0,0,0,0,0,19;
1000,2120,1740,560,0,160;
378,2352,2016,1554,84,0,0,0,0,0,0,0,0,0,0,0,0,0,21;
1056,3828,2310,308,176,220;
460,2852,4117,2530,644,138,0,0,0,0,0,0,0,0,0,0,0,0,0,0,23;
2352,5856,2376,288;
550,4550,4800,3350,900,250,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25;
1950,6708,5772,1612,650,208;
648,6372,6399,4968,432,810,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,27;
2380,10024,5712,3584,616,224;
754,8642,8497,6148,783,928,0,870,232,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,29;
6330,14640,5340,2340;
868,11532,11191,7378,806,1302,0,496,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,31;
3712,16384,12000,6016,1248;
990,14454,14619,12540,2079,1782,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,33;

Scott R. Shannon, Image for n=4 with edge-count coloring. This is the first n-gon with 3-gon to 3-gon contacts.
Scott R. Shannon, Image for n=6 with edge-count coloring. This is the first n-gon with 3-gon to 4-gon contacts.
Scott R. Shannon, Image for n=7 with edge-count coloring. This is the first n-gon with 3-gon to 5-gon contacts, other than n=5 which has an internal 5-gon surrounded by 3-gons by its construction.
Scott R. Shannon, Image for n=9 with edge-count coloring. This is the first n-gon with 3-gon to 6-gon contacts.
Scott R. Shannon, Image for n=15 with edge-count coloring. This is the first n-gon with 3-gon to 7-gon contacts, other than n=7 which has an internal 7-gon surrounded by 3-gons by its construction.
Scott R. Shannon, Image for n=15 with random distance-based coloring.
Scott R. Shannon, Image for n=13 with edge-count coloring. This is the first n-gon with 3-gon to 8-gon contacts.
Scott R. Shannon, Image for n=29 with edge-count coloring. This is the first n-gon with 3-gon to 10-gon and 3-gon to 11-gon contacts.

Cf. A335614 (4-gon contacts), A335646 (5-gon contacts), A337330 (6-gon contacts), A007678, A135565, A007569, A062361, A331450, A331451.

For odd n, with n > 3, row(n,n) = n as the n-gon contains a smaller n-gon at its center surrounded by 3-gons.

A335614 Irregular table read by rows: row n gives the number of 4-gon to k-gon contacts, with k>=4, for a regular n-gon with all diagonals drawn.

Original entry on oeis.org

0, 0, 0, 6, 0, 5, 24, 18, 36, 18, 90, 20, 110, 110, 44, 168, 208, 208, 52, 378, 140, 570, 330, 330, 640, 320, 32, 918, 646, 238, 34, 34, 990, 1254, 1330, 304, 38, 76, 1540, 1040, 80, 2478, 1722, 1008, 42, 2948, 1408, 44, 132, 132, 2852, 2254, 1012, 92, 46, 3624, 1536, 144

Offset: 3

Scott R. Shannon, Aug 23 2020

See A333654 for the number of 3-gon to k-gon contacts, with k>=3.
See A335646 for the number of 5-gon to k-gon contacts, with k>=5.
See A337330 for the number of 6-gon to k-gon contacts, with k>=6.
See A007678 for the number of regions and images of other n-gons.

			The table begins:
.
0;
0;
0;
6;
0,5;
24;
18,36,18;
90,20;
110,110,44;
168;
208,208,52;
378,140;
570,330,330;
640,320,32;
918,646,238,34,34;
990;
1254,1330,304,38,76;
1540,1040,80;
2478,1722,1008,42;
2948,1408,44,132,132;
2852,2254,1012,92,46;
3624,1536,144;
4650,3600,1450,150,150;
4784,3380,572,52;
6696,4968,2160,54,54;
7280,3752,2072,168;
8990,7308,3016,0,232,0,0,58;
7770,1200,300;
13020,9920,3472,186,434,0,124;
12160,8576,2048,256;
14784,11418,5742,792,330;

Scott R. Shannon, Image for n=6 with edge-number coloring. This is the first n-gon with 4-gon to 4-gon contacts.
Scott R. Shannon, Image for n=7 with edge-number coloring. This is the first n-gon with 4-gon to 5-gon contacts.
Scott R. Shannon, Image for n=9 with edge-number coloring. This is the first n-gon with 4-gon to 6-gon contacts.
Scott R. Shannon, Image for n=17 with edge-number coloring. This is the first n-gon with 4-gon to 7-gon and 4-gon to 8-gon contacts.
Scott R. Shannon, Image for n=31 with edge-number coloring. This is the first n-gon with 4-gon to 10-gon contacts.
Scott R. Shannon, Image for n=31 with random distance-based coloring.
Scott R. Shannon, Image for n=29 with edge-number coloring. This is the first n-gon with 4-gon to 11-gon contacts.
Scott R. Shannon, Image for n=18 with edge-number coloring. This is the only n-gon from n=13 to n=33 that has only 4-gon to 3-gon and 4-gon to 4-gon contacts. It is likely the last n-gon with this property.
Scott R. Shannon, Image for n=18 with random distance-based coloring.

Cf. A333654 (3-gon contacts), A335646 (5-gon contacts), , A337330 (6-gon contacts), A007678, A135565, A007569, A062361, A331450, A331451.

cp's OEIS Frontend

A007678 Number of regions in regular n-gon with all diagonals 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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A331449 a(0) = 1 by convention; for n>0, a(n) is the number of points in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A358783 The number of edges formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A342222 a(n) is the smallest m such that a regular m-gon with all diagonals drawn contains a cell with n sides, or a(n) = -1 if no such m exists.

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A331448 a(0) = 0 by convention; for n>0, a(n) is the number of edges in the n-th figure shown in A255011 (meaning the figure with 4n points on the perimeter).

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A371375 Place n equally spaced points around the circumference of a circle and then, for each pair of points, draw two distinct circles, whose radii are the same as the first circle, such that both points lie on their circumferences. The sequence gives the total number of (curved) edges formed.

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A359047 Number of distinct edges among all circles that can be constructed on vertices of an n-sided regular polygon, using only a compass.

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A333654 Irregular table read by rows: row n gives the number of 3-gon to k-gon contacts, with k>=3, for a regular n-gon with all diagonals drawn.

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