A350000 -id:A350000 - 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

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

A349967 Number of vertices in regular n-gon after 2 generations of mitosis.

Original entry on oeis.org

3, 5, 15, 25, 119, 81, 504, 311, 1309, 481, 3601, 1639, 5985, 3329, 12070, 4033, 19418, 11261, 26019, 18107, 42872, 16153, 60900, 38897, 83970, 54993, 137460, 38161, 159650, 97089, 204930, 136307, 262010, 144361, 337810, 223327, 404508, 292241, 529310, 243685, 662071, 441145, 749385

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Dec 07 2021

nonn

See A350000 for further details.

Cf. A349807 (cells), A349968 (edges), A350000, A007569.

A349968 Number of edges in regular n-gon after 2 generations of mitosis.

Original entry on oeis.org

3, 8, 35, 66, 308, 232, 1305, 900, 3399, 1428, 9061, 4704, 15345, 9424, 30294, 11376, 48773, 30840, 66738, 49148, 108330, 46320, 153825, 105690, 212355, 148876, 337328, 108330, 399404, 264320, 514866, 365874, 657265, 393264, 844969, 598272, 1017510, 774520, 1319257, 670152, 1641224, 1171192

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Dec 07 2021

nonn

See A350000 for further details.

Cf. A349807 (cells), A349967 (vertices), A350000, A135565.

A349807 Number of cells in regular n-gon after 2 generations of mitosis.

Original entry on oeis.org

1, 4, 21, 42, 190, 152, 802, 590, 2091, 948, 5461, 3066, 9361, 6096, 18225, 7344, 29356, 19580, 40720, 31042, 65459, 30168, 92926, 66794, 128386, 93884, 199869, 70170, 239755, 167232, 309937, 229568, 395256, 248904, 507160, 374946, 613003, 482280, 789948, 426468, 979154, 730048, 1129816

Offset: 3

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

nonn

We would very much like to have a formula for a(n) similar to the Poonen-Rubinstein formula for column 1 of A350000 (cf. A007678).

Cf. A349967 (vertices), A349968 (edges), A007678.

This is the k=2 column of the array A350000.

A350501 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

3, 3, 4, 3, 5, 5, 3, 5, 10, 6, 3, 5, 15, 19, 7, 3, 5, 20, 25, 42, 8, 3, 5, 25, 25, 119, 57, 9, 3, 5, 30, 25, 231, 81, 135, 10, 3, 5, 35, 25, 378, 81, 504, 171, 11, 3, 5, 40, 25, 560, 81, 1017, 311, 341, 12, 3, 5, 45, 25, 777, 81, 1620, 361, 1309, 313, 13

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 01 2022

See A350000 for further details and images of the n-gons.

			The table begins:
.
      |               Number of vertices after k generations
  n\k |  0,    1,     2,     3,     4,      5,      6,      7,      8,      9, ...
----------------------------------------------------------------------------------
   3  |  3,    3,     3,     3,     3,      3,      3,      3,      3,      3, ...
   4  |  4,    5,     5,     5,     5,      5,      5,      5,      5,      5, ...
   5  |  5,   10,    15,    20,    25,     30,     35,     40,     45,     50, ...
   6  |  6,   19,    25,    25,    25,     25,     25,     25,     25,     25, ...
   7  |  7,   42,   119,   231,   378,    560,    777,   1029,   1316,   1638, ...
   8  |  8,   57,    81,    81,    81,     81,     81,     81,     81,     81, ...
   9  |  9,  135,   504,  1017,  1620,   2313,   3096,   3969,   4932,   5985, ...
  10  | 10,  171,   311,   361,   411,    461,    511,    561,    611,    661, ...
  11  | 11,  341,  1309,  2629,  4169,   5929,   7909,  10109,  12529,   1516, ...
  12  | 12,  313,   481,   481,   481,    481,    481,    481,    481,    481, ...
  13  | 13,  728,  3601,  8125, 13624,  20098,  27547,  35971,  45370,  55744, ...
  14  | 14,  771,  1639,  2129,  2619,   3109,   3599,   4089,   4579,   5069, ...
  15  | 15, 1380,  5985, 13125, 22185,  32970,  45480,  59715,  75675,  93360, ...
  16  | 16, 1393,  3329,  4257,  4897,   5537,   6177,   6817,   7457,   8097, ...
  17  | 17, 2397, 12070, 28628, 50558,  77758, 110228, 147968, 190978, 239258, ...
  18  | 18, 1855,  4033,  5815,  7363,   8713,  10063,  11413,  12763,  14113, ...
  19  | 19, 3895, 19418, 44992, 77786, 117800, 165034, 219488, 281162, 350056, ...
  20  | 20, 3861, 11261, 16641, 20741,  24841,  28941,  33041,  37141,  41241, ...
  21  | 21, 6006, 26019, 55734, 92484, 136269, 187089, 244944, 309834, 381759, ...
  22  | 22, 5963, 18107, 27413, 34343,  41273,  48203,  55133,  62063,  68993, ...
.

Scott R. Shannon, Illustration of T(10,1).
Scott R. Shannon, Illustration of T(10,2).
Scott R. Shannon, Illustration of T(10,3).
Scott R. Shannon, Illustration of T(11,1).
Scott R. Shannon, Illustration of T(11,2).
Scott R. Shannon, Illustration of T(11,3).

Cf. A350000 (n-gons), A350502 (edges), A007569 (column 1), A349967 (column 2), A331450, A349968.

A352533 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex is fully connected to all other vertices in counterclockwise order before the next vertex, in counterclockwise order, is chosen.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 3, 5, 8, 11, 1, 2, 3, 4, 6, 9, 13, 16, 20, 24, 1, 2, 3, 4, 5, 7, 10, 14, 19, 22, 27, 34, 38, 45, 50, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 29, 34, 39, 46, 50, 56, 62, 67, 74, 80, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 37, 42, 49, 58, 69, 73, 80, 90, 103, 108, 117, 130, 136, 147, 154

Offset: 3

Scott R. Shannon, Mar 19 2022

To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to all other vertices where those vertices are chosen in a counterclockwise direction around the n-gon from the starting vertex. Once all those diagonals are drawn the next neighboring vertex, in a counterclockwise direction from the starting vertex, is chosen. This vertex is then connected to all other vertices in the same way. This method is repeated until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

			The table begins:
1;
1,2,4;
1,2,3,5,8,11;
1,2,3,4,6,9,13,16,20,24;
1,2,3,4,5,7,10,14,19,22,27,34,38,45,50;
1,2,3,4,5,6,8,11,15,20,26,29,34,39,46,50,56,62,67,74,80;
1,2,3,4,5,6,7,9,12,16,21,27,34,37,42,49,58,69,73,80,90,103,108,117,130,136,147,154;
1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,46,51,58,65,75,86,90,97,107,116,130,135,143, \
      154,164,170,180,194,201,212,220;
1,2,3,4,5,6,7,8,9,11,14,18,23,29,36,44,53,56,61,68,77,88,101,116,120,127,137,150, \
      166,185,190,199,212,229,250,256,267,283,304,311,324,343,351,366,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 7-gon.

Scott R. Shannon, Table for n=3..100.
Scott R. Shannon, Image for the 7-gon after first vertex is connected to all other vertices. Total regions = 5.
Scott R. Shannon, Image for the 7-gon after second vertex is connected to all other vertices. Total regions = 19.
Scott R. Shannon, Image for the 7-gon after third vertex is connected to all other vertices. Total regions = 34.
Scott R. Shannon, Image for the 7-gon after fourth vertex is connected to all other vertices. Total regions = 45.
Scott R. Shannon, Image for the 7-gon after fifth vertex is connected to all other vertices. Total regions = 50.

Cf. A007678, A350000, A344857.

The last term in each row n = A007678(n).

A349808 Number of cells in a regular 7-gon after n generations of mitosis.

Original entry on oeis.org

1, 50, 190, 400, 680, 1030, 1450, 1940, 2500, 3130, 3830, 4600, 5440, 6350, 7330, 8380, 9500, 10690, 11950, 13280, 14680, 16150, 17690, 19300, 20980, 22730, 24550, 26440, 28400, 30430, 32530, 34700, 36940, 39250, 41630, 44080, 46600, 49190, 51850, 54580, 57380, 60250, 63190, 66200, 69280, 72430, 75650, 78940, 82300, 85730, 89230, 92800

Offset: 0

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

nonn

See A350000 for background information.

Scott R. Shannon, Illustration for a(1) = 50, the 7-gon after the first splitting.
Scott R. Shannon, Illustration for a(2) = 190, the 7-gon after the second splitting.
Scott R. Shannon, Illustration for a(3) = 400 (third generation).
Scott R. Shannon, Illustration for a(4) = 680 (fourth generation).
Scott R. Shannon, Illustration for a(5) = 1030 (fifth generation).
Index entries for linear recurrences with constant coefficients, signature (3, -3, 1).

Row 7 of the array in A350000.

with(LinearAlgebra):
M:=Matrix(5,5,[ [1,4,10,18,35], [0,0,0,6,7], [0,0,1,0,7], [0,0,0,0,0], [0,0,0,0,1]]);
v:=Matrix(5,1,[ [0], [0], [0], [0], [1]]); u:=Matrix(1,5,[1,1,1,1,1]);
A349808:=n->(u.M^n.v)[1,1];

a(0)=1; for n>=1, a(n) = 35*k^2+35*k-20.
a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n >= 4, with a(0) = 1, a(1) = 50, a(2) = 190, a(3) = 400.
G.f.: (21*x^3-43*x^2-47*x-1)/(x-1)^3.
This sequence is most easily analyzed via the transition matrix M described in the Maple program.

A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

Original entry on oeis.org

3, 3, 4, 3, 8, 5, 3, 8, 20, 6, 3, 8, 35, 42, 7, 3, 8, 50, 66, 91, 8, 3, 8, 65, 66, 308, 136, 9, 3, 8, 80, 66, 630, 232, 288, 10, 3, 8, 95, 66, 1057, 232, 1305, 390, 11, 3, 8, 110, 66, 1589, 232, 2808, 900, 715, 12, 3, 8, 125, 66, 2226, 232, 4581, 1050, 3399, 756, 13

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jan 01 2022

See A350000 for further details and images of the n-gons.

			The table begins:
.
      |               Number of edges after k generations
  n\k |  0,     1,     2,      3,      4,      5,      6,      7,      8, ...
----------------------------------------------------------------------------------
   3  |  3,     3,     3,      3,      3,      3,      3,      3,      3, ...
   4  |  4,     8,     8,      8,      8,      8,      8,      8,      8, ...
   5  |  5,    20,    35,     50,     65,     80,     95,    110,    125, ...
   6  |  6,    42,    66,     66,     66,     66,     66,     66,     66, ...
   7  |  7,    91,   308,    630,   1057,   1589,   2226,   2968,   3815, ...
   8  |  8,   136,   232,    232,    232,    232,    232,    232,    232, ...
   9  |  9,   288,  1305,   2808,   4581,   6624,   8937,  11520,  14373, ...
  10  | 10,   390,   900,   1050,   1200,   1350,   1500,   1650,   1800, ...
  11  | 11,   715,  3399,   7271,  11803,  16995,  22847,  29359,  36531, ...
  12  | 12,   756,  1428,   1428,   1428,   1428,   1428,   1428,   1428, ...
  13  | 13,  1508,  9061,  22243,  38350,  57382,  79339, 104221, 132028, ...
  14  | 14,  1722,  4704,   6174,   7644,   9114,  10584,  12054,  13524, ...
  15  | 15,  2835, 15345,  35880,  62370,  94035, 130875, 172890, 220080, ...
  16  | 16,  3088,  9424,  12496,  14416,  16336,  18256,  20176,  22096, ...
  17  | 17,  4896, 30294,  77758, 141440, 220932, 316234, 427346, 554268, ...
  18  | 18,  4320, 11376,  16686,  21528,  25578,  29628,  33678,  37728, ...
  19  | 19,  7923, 48773, 122607, 218101, 335255, 474069, 634543, 816677, ...
  20  | 20,  8360, 30840,  48260,  60560,  72860,  85160,  97460, 109760, ...
  21  | 21, 12180, 66738, 153069, 260505, 389046, 538692, 709443, 901299, ...
  22  | 22, 12782, 49148,  79442, 100232, 121022, 141812, 162602, 183392, ...
.

Cf. A350000 (n-gons), A350501 (vertices), A135565 (column 1), A349968 (column 2), A331450, A349967.

A352866 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex in turn is connected to the vertex two to its left, then three to its left, then four... until all vertices are connected by diagonals.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 4, 6, 8, 11, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 1, 2, 4, 6, 8, 10, 12, 15, 18, 22, 27, 32, 37, 43, 50, 1, 2, 4, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 39, 44, 50, 57, 62, 68, 74, 80, 1, 2, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 31, 36, 41, 46, 51, 57, 64, 71, 79, 88, 98, 108, 118, 129, 141, 154

Offset: 3

Scott R. Shannon, Apr 06 2022

To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to the vertex two to its left. The left neighboring vertex of the starting vertex is then chosen and this is connected to the vertex two to its left. This process is continued until all vertices are connected by diagonals to the vertices two to their left. The initial vertex is then chosen again and it is connected to the vertex three to its left. Its left neighboring vertex is then connected to the vertex three to its left, and so on. This process of connecting all vertices to those on their left by diagonals, where the step size increases by one after each complete circuit of the n-gon, is continued until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

			The table begins:
1;
1,2,4;
1,2,4,6,8,11;
1,2,4,6,8,10,13,16,20,24;
1,2,4,6,8,10,12,15,18,22,27,32,37,43,50;
1,2,4,6,8,10,12,14,17,20,24,29,34,39,44,50,57,62,68,74,80;
1,2,4,6,8,10,12,14,16,19,22,26,31,36,41,46,51,57,64,71,79,88,98,108,118,129, \
        141,154;
1,2,4,6,8,10,12,14,16,18,21,24,28,33,38,43,48,53,58,64,71,78,86,95,105,115, \
        125,135,146,158,171,180,190,200,210,220;
1,2,4,6,8,10,12,14,16,18,20,23,26,30,35,40,45,50,55,60,65,71,78,85,93,102,112, \
        122,132,142,152,163,175,188,201,215,230,246,263,280,297,315,334,354,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 10-gon.

Scott R. Shannon, Table for n=3..100.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices two to their left. Total regions = 21.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices three to their left. Total regions = 71.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices four to their left. Total regions = 171.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices five to their left. Total regions = 220.

Cf. A352533, A007678, A350000, A344857.

The last term in each row n = A007678(n).

cp's OEIS Frontend

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

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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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A349967 Number of vertices in regular n-gon after 2 generations of mitosis.

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A349968 Number of edges in regular n-gon after 2 generations of mitosis.

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A349807 Number of cells in regular n-gon after 2 generations of mitosis.

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A350501 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in a regular n-gon after k generations of mitosis.

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A349808 Number of cells in a regular 7-gon after n generations of mitosis.

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A350502 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges in a regular n-gon after k generations of mitosis.

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