A007678 -id:A007678 - cp's OEIS frontend

A103314 Total number of subsets of the n-th roots of 1 that add to zero.

1, 1, 2, 2, 4, 2, 10, 2, 16, 8, 34, 2, 100, 2, 130, 38, 256, 2, 1000, 2, 1156, 134, 2050, 2, 10000, 32, 8194, 512, 16900, 2, 146854, 2, 65536, 2054, 131074, 158, 1000000, 2, 524290, 8198, 1336336, 2, 11680390, 2, 4202500, 54872, 8388610, 2, 100000000, 128

Offset: 0

Wouter Meeussen, Mar 11 2005

The term a(0) = 1 counts the single zero-sum subset of the (by convention) empty set of zeroth roots of 1.
I am inclined to believe that if S is a zero-sum subset of the n-th roots of 1, that n can be built up from (zero-sum) cyclically balanced subsets via the following operations: 1. A U B, where A and B are disjoint. 2. A - B, where B is a subset of A. - David W. Wilson, May 19 2005
Lam and Leung's paper, though interesting, does not apply directly to this sequence because it allows repetitions of the roots in the sums.
Observe that 2^n=a(n) (mod n). Sequence A107847 is the quotient (2^n-a(n))/n. - T. D. Noe, May 25 2005
From Max Alekseyev, Jan 31 2008: (Start)
Every subset of the set U(n) = { 1=r^0, r^1, ..., r^(n-1) } of the n-th power roots of 1 (where r is a fixed primitive root) defines a binary word w of the length n where the j-th bit is 1 iff the root r^j is included in the subset.
If d is the period of w with respect to cyclic rotations (thus d|n) then the periodic part of w uniquely defines some binary Lyndon word of the length d (see A001037). In turn, each binary Lyndon word of the length d, where dThe binary Lyndon words of the length n are different in this respect: only some of them correspond to n distinct zero-sum subsets of U(n) while the others do not correspond to such subsets at all. A110981(n) gives the number of binary Lyndon words of the length n that correspond to zero-sum subsets of U(n). (End)

Max Alekseyev and M. F. Hasler, Table of n, a(n) for n = 0..164
T. Y. Lam and K. H. Leung, On vanishing sums for roots of unity, arXiv:math/9511209 [math.NT], 1995.
T. D. Noe, Sums of Roots of Unity Plots
T. D. Noe, Proof a(n)=a(s(n))^(n/ s(n))
Sasha Rybak, Zero sums of roots of unity (in Russian), forum dxdy.ru.
Gary Sivek, On vanishing sums of distinct roots of unity, #A31, Integers 10 (2010), 365-368.

Equals A070894 + 1. A107847(n) = (2^n - A103314(n))/n, A110981 = A001037 - A107847.
Row sums of A103306. See also A006533, A006561, A006600, A007569, A007678.
Cf. A070925, A107753 (number of primitive subsets of the n-th roots of unity summing to zero), A107754 (number of subsets of the n-th roots of unity summing to one), A107861 (number of distinct values in the sums of all subsets of the n-th roots of unity).
Cf. A322366.

Needs["DiscreteMath`Combinatorica`"]; Table[Plus@@Table[Count[ (KSubsets[ Range[n], k]), q_List/;Chop[ Abs[Plus@@(E^(2.*Pi*I*q/n))]]==0], {k, 0, n}], {n, 15}] (* T. D. Noe *)

/* This program implements all known results; it works for all n except for 165, 195, 210, 231, 255, 273, 285, 330, 345, ... */
A103314(n) = { local(f=factor(n)); n<2 & return(1); n==f[1,1] & return(2);
vecmax(f[,2])>1 & return(A103314(f=prod(i=1,#f~,f[i,1]))^(n/f));
if( 2==#f=f[,1], return(2^f[1]+2^f[2]-2));
#f==3 & f[1]==2 & return(sum(j=0,f[2],binomial(f[2],j)*(2^j+2^(f[2]-j))^f[3])
+(2^f[2]+2)^f[3]+(2^f[3]+2)^f[2]-2*((2^f[2]+1)^f[3]+(2^f[3]+1)^f[2])+2^(f[2]*f[3]));
n==105 & return(166093023482); error("A103314(n) is unknown for n=",n) }
/* Max Alekseyev and M. F. Hasler, Jan 31 2008 */

a(n) = A070894(n)+1.
a(2^n) = 2^(2^(n-1)). - Dan Asimov and Gareth McCaughan, Mar 11 2005
a(2n) = a(n)^2 if n is even. If p, q are primes, a(pq) = 2^p+2^q-2. In particular, if p is prime, a(2p) = 2^p + 2. - Gareth McCaughan, Mar 12 2005
a(n) == 2^n (mod n), a(p) = 2 (p prime). - David W. Wilson, May 08 2005
It appears that a(n) = a(s(n))^(n/s(n)) where s(n) = A007947(n) is the squarefree kernel of n. This is true if all zero-sum subsets of the n-th roots of 1 are formed by set operations on cyclic subsets. If true, A103314 is determined by its values on squarefree numbers (A005117). Some consequences would be a(p^n) = 2^p^(n-1), a(p^m q^n) = (2^p+2^q+2)^(p^(m-1) q^(n-1)) and a(p^2 n) = a(pn)^p for primes p and q. - David W. Wilson, May 08 2005
a(pn) = a(n)^p when p is prime and p|n; a(2p) = 2^p+2 when p is an odd prime. More generally a(pq) = 2^p+2^q-2 when p, q are distinct primes. - Gareth McCaughan, Mar 12 2005
For distinct odd primes p and q, a(2pq) = (2^p+2)^q + (2^q+2)^p - 2(2^p+1)^q - 2(2^q+1)^p + 2^(pq) + SUM[j=0..p] binomial(p,j)(2^j+2^(p-j))^q. - Sasha Rybak, Sep 21 2007.
a(n) = n*A110981(n) + 2^n - n*A001037(n). - Max Alekseyev, Jan 14 2008

More terms from David W. Wilson, Mar 12 2005
Scott Huddleston (scotth(AT)ichips.intel.com) finds that a(30) >= 146854 and conjectures that is the true value of a(30). - Mar 24 2005. Confirmed by Meeussen and Wilson.
More terms from T. D. Noe, May 25 2005
Further terms from Max Alekseyev and M. F. Hasler, Jan 07 2008
Edited by M. F. Hasler, Feb 06 2008
Duplicate Mathematica program deleted by Harvey P. Dale, Jun 28 2021

A006522 4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.

Original entry on oeis.org

1, 0, 0, 1, 4, 11, 25, 50, 91, 154, 246, 375, 550, 781, 1079, 1456, 1925, 2500, 3196, 4029, 5016, 6175, 7525, 9086, 10879, 12926, 15250, 17875, 20826, 24129, 27811, 31900, 36425, 41416, 46904, 52921, 59500, 66675, 74481, 82954, 92131

Offset: 0

N. J. A. Sloane

Let A be the Hessenberg matrix of order n, defined by: A[1,j]=A[i,i]:=1, A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=5, a(n)=coeff(charpoly(A,x),x^(n-4)). - Milan Janjic, Jan 24 2010
From Ant King, Sep 14 2011: (Start)
Consider the array formed by the polygonal numbers of increasing rank A139600
  0, 1, 3,  6, 10, 15, 21,  28,  36,  45, ...  A000217(n)
  0, 1, 4,  9, 16, 25, 36,  49,  64,  81, ...  A000290(n)
  0, 1, 5, 12, 22, 35, 51,  70,  92, 117, ...  A000326(n)
  0, 1, 6, 15, 28, 45, 66,  91, 120, 153, ...  A000384(n)
  0, 1, 7, 18, 34, 55, 81, 112, 148, 189, ...  A000566(n)
  0, 1, 8, 21, 40, 65, 96, 133, 176, 225, ...  A000567(n)
  ...
Then, for n>=2, a(n) is the diagonal sum of this polygonal grid.
(End)
Binomial transform of (1, -1, 1, 0, 1, 0, 0, 0, ...). - Gary W. Adamson, Aug 26 2015

			For a pentagon in general position, 11 regions are formed (Comtet, Fig. 20, p. 74).

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 8.
Ross Honsberger, Mathematical Gems, M.A.A., 1973, p. 102.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. W. Freeman, The number of regions determined by a convex polygon, Math. Mag., 49 (1976), 23-25.
Math Forum, Regions of a circle Cut by Chords to n points.
V. Meally, Letter to N. J. A. Sloane, May 1975.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Eric Weisstein's World of Mathematics, Polygon Diagonal.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Partial sums of A004006.

Cf. A000332, A000217.

[Binomial(n, 4)+Binomial(n-1, 2): n in [0..40]]; // Vincenzo Librandi, Jun 09 2013

A006522 := n->(1/24)*(n-1)*(n-2)*(n^2-3*n+12):
seq(A006522(n), n=0..40);
A006522:=-(1-z+z**2)/(z-1)**5; # Simon Plouffe in his 1992 dissertation; gives sequence except for three leading terms

a=2;b=3;s=4;lst={1,0,0,1,s};Do[a+=n;b+=a;s+=b;AppendTo[lst,s],{n,2,6!,1}];lst (* Vladimir Joseph Stephan Orlovsky, May 24 2009 *)
Table[Binomial[n,4]+Binomial[n-1,2],{n,0,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{1,0,0,1,4},40] (* Harvey P. Dale, Jul 11 2011 *)
CoefficientList[Series[-(((x - 1) x (x (4 x - 5) + 5) + 1) / (x - 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 09 2013 *)
a[n_] := If[n==0, 1, Sum[PolygonalNumber[n-k+1, k], {k, 0, n-2}]];
a /@ Range[0, 40] (* Jean-François Alcover, Jan 21 2020 *)

a(n)=1/24*n^4 - 1/4*n^3 + 23/24*n^2 - 7/4*n + 1 \\ Charles R Greathouse IV, Feb 09 2017

a(n) = binomial(n,4) + binomial(n-1,2) = A000332(n) + A000217(n-2).
a(n) = binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4). - Zerinvary Lajos, Jul 23 2006
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5); a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=4. - Harvey P. Dale, Jul 11 2011
G.f.: -((x-1)*x*(x*(4*x-5)+5)+1)/(x-1)^5. - Harvey P. Dale, Jul 11 2011
a(n) = (n^4 - 6*n^3 + 23*n^2 - 42*n + 24)/24. - T. D. Noe, Oct 16 2013
For odd n, a(n) = A007678(n). - R. J. Mathar, Nov 22 2017
a(n) = a(3-n) for all n in Z. - Michael Somos, Nov 23 2021
Sum_{n>=3} 1/a(n) = 66/25 - (4/5)*sqrt(3/13)*Pi*tanh(sqrt(39)*Pi/2). - Amiram Eldar, Aug 23 2022

A006600 Total number of triangles visible in regular n-gon with all diagonals drawn.

Original entry on oeis.org

1, 8, 35, 110, 287, 632, 1302, 2400, 4257, 6956, 11297, 17234, 25935, 37424, 53516, 73404, 101745, 136200, 181279, 236258, 306383, 389264, 495650, 620048, 772785, 951384, 1167453, 1410350, 1716191, 2058848, 2463384, 2924000, 3462305, 4067028, 4776219, 5568786, 6479551

Offset: 3

N. J. A. Sloane

Place n equally-spaced points on a circle, join them in all possible ways; how many triangles can be seen?

			a(4) = 8 because in a quadrilateral the diagonals cross to make four triangles, which pair up to make four more.

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=3..1000
Sascha Kurz, m-gons in regular n-gons
Victor Meally, Letter to N. J. A. Sloane, no date.
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 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,...
T. Sillke, Number of triangles for convex n-gon
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.
Sequences formed by drawing all diagonals in regular polygon

Often confused with A005732.
Cf. A203016, A260417.
Row sums of A363174.
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]; Tri[n_]:=n(n-1)(n-2)(n^3+18n^2-43n+60)/720 - del[2,n](n-2)(n-7)n/8 - del[4,n](3n/4) - del[6,n](18n-106)n/3 + del[12,n]*33n + del[18,n]*36n + del[24,n]*24n - del[30,n]*96n - del[42,n]*72n - del[60,n]*264n - del[84,n]*96n - del[90,n]*48n - del[120,n]*96n - del[210,n]*48n; Table[Tri[n], {n,3,1000}] (* T. D. Noe, Dec 21 2006 *)

a(2n-1) = A005732(2n-1) for n > 1; a(2n) = A005732(2n) - A260417(n) for n > 1. - Jonathan Sondow, Jul 25 2015

a(3)-a(8) computed by Victor Meally (personal communication to N. J. A. Sloane, circa 1975); later terms and recurrence from S. Sommars and T. Sommars.

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

A331931 The number of regions inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, 824712, 1183704, 1580868, 2067162, 2770476, 3585582, 4397172, 5665818, 6827736, 8318976, 10209948, 12364098, 14395164, 17194230, 20216808, 23436612, 27124416, 31817676, 35516328

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

Scott R. Shannon, Hexagon regions for n = 1.
Scott R. Shannon, Hexagon regions for n = 2.
Scott R. Shannon, Hexagon regions for n = 3.
Scott R. Shannon, Hexagon regions for n = 4.
Scott R. Shannon, Hexagon regions for n = 5.
Scott R. Shannon, Hexagon regions for n = 6.
Scott R. Shannon, Hexagon regions for n = 7.
Scott R. Shannon, Hexagon regions for n = 8.
Scott R. Shannon, Hexagon regions for n = 5, with random distance-based coloring.
Scott R. Shannon, Hexagon regions for n = 6, with random distance-based coloring.
Wikipedia, Hexagon.

Cf. A331932 (n-gons), A330845 (edges), A330846 (vertices), A007678, A092867, A331452, A331929.

a(9)-a(30) from Lars Blomberg, May 12 2020

A331929 The number of regions inside a pentagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

Original entry on oeis.org

11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, 410860, 577721, 779340, 1035676, 1345030, 1730696, 2176040, 2724036, 3345880, 4087656, 4933200, 5921991, 7018210, 8300896, 9723300, 11339151, 13122120, 15150271, 17345140, 19843056

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 01 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.

Lars Blomberg, Table of n, a(n) for n = 1..35
Scott R. Shannon, Pentagon regions for n = 1.
Scott R. Shannon, Pentagon regions for n = 2.
Scott R. Shannon, Pentagon regions for n = 3.
Scott R. Shannon, Pentagon regions for n = 4.
Scott R. Shannon, Pentagon regions for n = 5.
Scott R. Shannon, Pentagon regions for n = 6.
Scott R. Shannon, Pentagon regions for n = 7.
Scott R. Shannon, Pentagon regions for n = 8.
Scott R. Shannon, Pentagon regions for n = 5, random distance-based coloring.
Scott R. Shannon, Pentagon regions for n = 6, random distance-based coloring
Wikipedia, Pentagon.

Cf. A331939 (n-gons), A329710 (edges), A330847 (vertices), A007678, A092867, A331452, A331931.

a(9) and beyond from Lars Blomberg, May 11 2020

A288187 Triangle read by rows: T(n,m) (n >= m >= 1) = number of chambers (or regions) formed by drawing the line segments connecting any two of the (n+1) X (m+1) lattice points in an n X m lattice polygon.

Original entry on oeis.org

4, 16, 56, 46, 176, 520, 104, 388, 1152, 2584, 214, 822, 2502, 5700, 12368, 380, 1452, 4392, 9944, 21504, 37400, 648, 2516, 7644, 17380, 37572, 65810, 115532, 1028, 3952, 12120, 27572, 59784, 105128, 184442, 294040, 1562, 6060, 18476, 42066, 91654, 161352, 282754, 450864, 690816

Offset: 1

Hugo Pfoertner, Jun 06 2017

Chambers are counted regardless of their numbers of vertices.

The n X m lattice polygon mentioned in the definition is an n X m grid of square cells, formed using a grid of n+1 X m+1 points. - N. J. A. Sloane, Feb 07 2019

			The diagonals of the 1 X 1 lattice polygon, i.e. the square, cut it into 4 triangles. Therefore T(1,1)=4.
Triangle begins
4,
16, 56,
46, 176, 520,
104, 388, 1152, 2584,
214, 822, 2502, 5700, 12368,
...

Lars Blomberg, Table of n, a(n) for n = 1..325 (The first 25 rows)
Lars Blomberg, Colored illustration for 3 x 3
Lars Blomberg, Colored illustration for 4 X 4
Lars Blomberg, Colored illustration for 5 X 3
Lars Blomberg, Colored illustration for 5 X 5
Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
Huntington Tracy Hall, Counterexamples in Discrete Geometry. Dissertation, Department of Mathematics, University of California Berkeley, Fall 2004.
Serkan Hosten, Diane Maclagan, Bernd Sturmfels, Supernormal Vector Configurations, arXiv:math/0105036 [math.CO], 4 May 2001.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004.
Marc E. Pfetsch, Günter M. Ziegler, Large Chambers in a Lattice Polygon (Notes), March 28, 2001, December 13, 2004. [Cached copy, with permission]
Hugo Pfoertner, Illustrations of Chamber Complexes up to 5 X 5.

Cf. A288177, A288180, A288181.
The first column is A306302. For column 2 see A333279, A333280, A333281.
If the initial points are arranged around a circle rather than a square we get A006533 and A007678.

T(4,1) added from A306302. - N. J. A. Sloane, Feb 07 2019

T(3,3) corrected and rows for n=4..9 added by Max Alekseyev, Apr 05 2019.

A358782 The number of regions 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

1, 7, 12, 66, 85, 281, 264, 802, 821, 1893, 1740, 3810, 3725, 6871, 6448, 11748, 11125, 18317, 17160, 27616, 26797, 40067, 37176, 56826, 54653, 77707, 74788, 103734, 101041, 136835, 131744, 176584, 172109, 223931, 216900, 281090, 273829, 348583, 337480, 425950, 416641

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.
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

Scott R. Shannon, Image for n = 2. In this and other images the points defining the circle diameters are show as white dots.
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 = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 23.

Cf. A358746 (vertices), A358783 (edges), A359009 (k-gons), A007678, A344857.

See allso A370976-A370979.

a(n) = A358783(n) - A358746(n) + 1 by Euler's formula.

Previous Showing 21-30 of 144 results. Next

This sequence and A007678 are two equivalent ways of presenting the same sequence. - N. J. A. Sloane, Jan 23 2020

In contrast to A007678, which only counts the polygons, this sequence also counts the n segments of the circle bounded by the arc of the circle and the straight line, both joining two neighboring points on the circle. Therefore a(n) = A007678(n) + n. - M. F. Hasler, Dec 12 2021

For odd n, a(n) is given by the equation in the Formula section below. See also A344866. For even n no such equation is currently known, although one similar to the general formula for the number of polygons inside an n-gon, see A007678, likely exists.
The number of open regions, those outside the polygons with unbounded area and two edges that go to infinity, for n >= 3 is given by n*(n-1) = A002378(n-1).
See A345025 for the total number of all areas, both polygons and open regions.

cp's OEIS Frontend

A006533 Place n equally-spaced points around a circle and join every pair of points by a chord; this divides the circle into a(n) regions.

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A344857 Number of polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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A103314 Total number of subsets of the n-th roots of 1 that add to zero.

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A006522 4-dimensional analog of centered polygonal numbers. Also number of regions created by sides and diagonals of a convex n-gon in general position.

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A006600 Total number of triangles visible 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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A331931 The number of regions inside a hexagon formed by the straight line segments mutually connecting all vertices and all points that divide the sides into n equal parts.

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