A006561 -id:A006561 - cp's OEIS frontend

A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.

1, 1, 3, 3, 7, 7, 14, 14, 25, 21, 41, 40, 63, 60, 92, 72, 129, 121, 175, 166, 231, 192, 298, 285, 377, 360, 469, 350, 575, 553, 696, 666, 833, 744, 987, 956, 1159, 1123, 1350, 1165, 1561, 1508, 1793, 1741, 2047, 1875, 2324, 2255, 2625, 2563, 2951, 2761, 3303, 3214, 3682, 3588, 4089, 3695

Offset: 3

Martin Renner, Jan 05 2013

			a(5) = 3 since the 11 regions of a regular pentagon with all diagonals drawn consist of three different noncongruent polygons: two different triangles (each 5 times) and 1 pentagon.
a(6) = 3 since the 24 regions of the regular hexagon with all diagonals drawn consist of three different noncongruent polygons: 2 triangles (one 6 times, one 12 times) and 1 quadrilateral (6 times).
a(7) = 7 since the 50 regions of the regular heptagon with all diagonals drawn consist of seven different noncongruent polygons: 4 triangles (three 7 times, one 14 times), 1 quadrilateral (7 times), 1 pentagon (7 times) and 1 heptagon.

Sascha Kurz, Anzahl von Dreiecken eines regelmäßigen n-Ecks.
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.
Index to sequences on drawing diagonals in regular polygons.

Cf. A006561, A007678, A165217, A187782.

Corrected a(12) and a(16), extended from a(18) through a(60), corrected small typo in a(7) example - Christopher Scussel, Jun 23 2023

A230281 The least possible number of intersection points of the diagonals in the interior of a convex n-gon with all diagonals drawn.

Original entry on oeis.org

0, 1, 5, 13, 29, 49

Offset: 3

Vladimir Letsko, Oct 15 2013

Perhaps a(9) = 94.
After removing two points from the regular 12-gon, that is, removing the corresponding points at 12 o'clock and 2 o'clock, there will be only 157 intersection points of the diagonals, it is less than 161, which is the number of intersections of diagonals in the interior of regular 10-gon. So, a(10) <= 157 < 161 = A006561(10). - Guang Zhou, Jul 27 2018
The greatest possible number of intersection points occurs when each set of four vertices gives diagonals with a unique intersection point.  Thus, a(n) <= binomial(n,4) = A000332(n). - Michael B. Porter, Jul 30 2018

			a(6) = 13 because the number of intersection points of the diagonals in the interior of convex hexagon is equal to 13 if 3 diagonals meet in one point, and this number cannot be less than 13 for any hexagon.

Nathaniel Johnston, Illustration of a(4), a(5), and a(6)
Vladimir Letsko, Mathematical Marathon at VSPU, Problem 102 (in Russian)
Vladimir Letsko, Illustration of a(8) = 49 (the regular octagon provides another example)
V. A. Letsko and M. A. Voronina, Classification of convex polygons, Grani Poznaniya, 1(11), 2011 (in Russian).
V. A. Letsko and M. A. Voronina, Illustration of a(7) = 29
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.

Cf. A000332, A006561, A160860.

A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

Original entry on oeis.org

1, 0, 0, 0, 4, 4, 0, 0, 10, 20, 5, 0, 20, 60, 30, 0, 35, 140, 105, 7, 56, 280, 280, 16, 84, 504, 630, 84, 120, 840, 1260, 180, 165, 1320, 2310, 462, 220, 1980, 3960, 796, 286, 2860, 6435, 1716, 364, 4004, 10010, 2856, 455, 5460, 15015, 5005, 560, 7280, 21840, 7744

Offset: 3

Paolo Xausa, May 19 2023

See Sommars and Sommars (1998) for a complete analysis of the problem.

			Array begins:
  n\k|     3     4     5     6
  ---+---------------------------
   3 |     1,    0,    0,    0;
   4 |     4,    4,    0,    0;
   5 |    10,   20,    5,    0;
   6 |    20,   60,   30,    0;
   7 |    35,  140,  105,    7;
   8 |    56,  280,  280,   16;
   9 |    84,  504,  630,   84;
  10 |   120,  840, 1260,  180;
  ...

Paolo Xausa, Table of n, a(n) for n = 3..10002 (rows 3..2502 of array, flattened).
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
Paolo Xausa, Illustration of T(9,6).
Sequences formed by drawing all diagonals in regular polygon

Cf. A000579, A006561, A006600 (row sums), A260417.

Cf. A000292 (column k = 3), A033488 (column k = 4), A174002 (column k = 5), A363173 (column k = 6).

A363174list[rowmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n,k]If[4<=k<=5,k,1]-If[k==6&&EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,rowmax},{k,3,6}]];A363174list[20]

T(n,3) = binomial(n,3) = A000292(n-2).
T(n,4) = 4*binomial(n,4) = A033488(n-3).
T(n,5) = 5*binomial(n,5) = A174002(n-4), for n >= 4.
T(n,6) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.
Sum_{k=3..6} T(n,k) = A006600(n).

A211382 Number of intersections of diagonals in the exterior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 7, 24, 63, 110, 242, 360, 650, 812, 1425, 1680, 2737, 2718, 4788, 5200, 7812, 8272, 12075, 11328, 17875, 18486, 25542, 26264, 35438, 29070, 47957, 48800, 63525, 64362, 82600, 77940, 105672, 106552, 133263, 134200, 165927, 149478, 204250, 205128, 248850, 249596, 300377

Offset: 3

Martin Renner, Feb 07 2013

nonn

Robin Visser, Table of n, a(n) for n = 3..95
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Cf. A006561, A146213, A211383.

def a(n):
    K = CyclotomicField(n); z = K.gen(); S = set()
    for i in range(n):
        for j in range(i+2, n):
            for k in range(j+1, n):
                for l in range(k+2, n+i):
                    x = (z^(i-j)-z^(j-i))*(z^l-z^k)-(z^(k-l)-z^(l-k))*(z^j-z^i)
                    y = (z^-j-z^-i)*(z^l-z^k)-(z^-l-z^-k)*(z^j-z^i)
                    if (not y.is_zero()): S.add(x/y)
    return len(S)  # Robin Visser, Jul 29 2024

a(n) = 1/24*n*(n-3)*(n-5)*(2*n-11) for n odd

a(17)-a(29) from Martin Renner, Feb 24 2013

More terms from Robin Visser, Jul 29 2024

A211383 Number of intersections of diagonals in the interior and exterior of a regular n-gon.

Original entry on oeis.org

0, 1, 5, 13, 42, 73, 189, 271, 572, 661, 1365, 1569, 2790, 3057, 5117, 4555, 8664, 9041, 13797, 14213, 20930, 18625, 30525, 30967, 43092, 43513, 59189, 45871, 79422, 79713, 104445, 104619, 134960, 124921, 171717, 171533, 215514, 215081, 267197, 234319, 327660, 326569, 397845, 396337

Offset: 3

Martin Renner, Feb 07 2013

nonn

Robin Visser, Table of n, a(n) for n = 3..95
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math/9508209 [math.MG], 1995-2006.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals

Cf. A006561, A146212, A211382.

def a(n):
    K = CyclotomicField(n); z = K.gen(); S = set()
    for i in range(n):
        for j in range(i+2, n):
            for k in range(j+1, n):
                for l in range(k+2, n+j):
                    x = (z^(i-j)-z^(j-i))*(z^l-z^k)-(z^(k-l)-z^(l-k))*(z^j-z^i)
                    y = (z^-j-z^-i)*(z^l-z^k)-(z^-l-z^-k)*(z^j-z^i)
                    if (l!=n+i) and (not y.is_zero()): S.add(x/y)
    return len(S)  # Robin Visser, Jul 29 2024

a(n) = (1/8)*n*(n-3)*(n^2-8*n+19) for n odd.

a(n) = A006561(n) + A211382(n).

More terms from Robin Visser, Jul 29 2024

A351924 The number of vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

Original entry on oeis.org

3, 5, 7, 11, 15, 21, 27, 29, 43, 53, 59, 75, 87, 85, 115, 131, 135, 165, 183, 185, 223, 245, 251, 291, 315, 317, 367, 395, 379, 453, 483, 485, 547, 581, 587, 651, 687, 689, 763, 803, 795, 885, 927, 925, 1015, 1061, 1067, 1155, 1203, 1205, 1303, 1355, 1359, 1461, 1515, 1517, 1627, 1685, 1659

Offset: 2

Scott R. Shannon, Feb 25 2022

nonn

No formula for a(n) is currently known.

			a(3) = 5 as a diagonal of a 6-gon, when all its vertices are connected by lines and where the diagonal passes through the center of the 6-gon, has five vertices on it - the two outer vertices, the central vertex, and two more vertices formed by intersections of the central diagonal with other diagonals. See the linked image of the 6-gon.

Scott R. Shannon, Image of the 6-gon.
Scott R. Shannon, Image of the 10-gon.
Scott R. Shannon, Image of the 18-gon.
Scott R. Shannon, Image of the 30-gon.

Cf. A007569, A006561, A146212.

A363173 Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.

Original entry on oeis.org

0, 0, 0, 0, 7, 16, 84, 180, 462, 796, 1716, 2856, 5005, 7744, 12376, 17508, 27132, 38160, 54264, 73788, 100947, 132216, 177100, 228748, 296010, 374808, 475020, 584140, 736281, 903168, 1107568, 1341232, 1623160, 1939308, 2324784, 2755380, 3262623, 3832080, 4496388

Offset: 3

Paolo Xausa, May 19 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 3..10000
Bjorn Poonen and Michael Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Steven E. Sommars and Tim Sommars, The Number of Triangles Formed by Intersecting Diagonals of a Regular Polygon, Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.5.
Paolo Xausa, Illustration of a(9).
Sequences formed by drawing all diagonals in regular polygon

Column k = 6 of A363174.

Cf. A000579, A006561, A260417.

A363173list[nmax_]:=Module[{d},d[m_,n_]:=Boole[Divisible[n,m]];Table[Binomial[n, 6]-If[EvenQ[n],((1/8n^2-9/8n+7/4)d[2,n]+3/4d[4,n]+(6n-106/3)d[6,n]-33d[12,n]-36d[18,n]-24d[24,n]+96d[30,n]+72d[42,n]+264d[60,n]+96d[84,n]+48d[90,n]+96d[120,n]+48d[210,n])n,0],{n,3,nmax}]];A363173list[50]

a(n) = binomial(n,6) = A000579(n) if n is odd, A000579(n) - A260417(n/2) if n is even.

A108053 Maximum number of diagonals of a regular n-gon that meet at a non-center point.

Original entry on oeis.org

0, 0, 2, 2, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2

Offset: 3

David W. Wilson, Jun 01 2005

Starting at a(13) = 2, sequence is periodic with period 30.

			In a 30-gon, there are non-center points where 7 diagonals meet, but no more than 7. Hence a(30) = 7.

Paolo Xausa, Table of n, a(n) for n = 3..10000
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, arXiv:math/9508209 [math.MG], 1995-2006.
Sequences formed by drawing all diagonals in regular polygon
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A006561, A007678.

LinearRecurrence[PadLeft[{1},30], {0, 0, 2, 2, 2, 3, 2, 3, 2, 4,2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 2, 5},120] (* Ray Chandler, Aug 27 2015 - adapted to new data by Paolo Xausa, May 15 2023 *)
PadRight[{0,0,2,2,2,3,2,3,2,4},120,{2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,5,2,3,2,3,2,7,2,3}] (* Harvey P. Dale, Jun 20 2021 - adapted to new data by Paolo Xausa, May 15 2023 *)

From Paolo Xausa, May 11 2023: (Start)
a(n) = 0 if n <= 4.
For n > 4:
a(n) = 2 if n is odd or n = 6;
a(n) = 3 if n != 6 is even but not divisible by 6;
a(n) = 4 if n = 12;
a(n) = 5 if n != 12 is divisible by 6 but not 30;
a(n) = 7 if n is divisible by 30. (End)

a(4), a(6) and a(12) corrected by Paolo Xausa, May 11 2023

A180686 Positive integers k such that the number of intersections of diagonals in the interior of a regular k-gon is prime.

Original entry on oeis.org

5, 6, 14, 24, 44, 58, 72, 76, 80, 84, 86, 104, 128, 134, 138, 180, 186, 188, 218, 228, 246, 256, 266, 280, 300, 320, 352, 360, 380, 390, 408, 450, 480, 508, 518, 524, 526, 532, 546, 548, 552, 576, 584, 590, 604, 616, 630, 656, 658, 686, 712, 724, 726, 728, 730

Offset: 1

Robert G. Wilson v, Sep 16 2010

nonn

Chris K. Caldwill & G. L. Honaker, Jr., Prime Curios!, The Dictionary of Prime Number Trivia, CreateSpace, Sept. 2009, p. 145.

Robin Visser, Table of n, a(n) for n = 1..1000

Cf. A006561.

del[m_, n_] := If[ Mod[n, m] == 0, 1, 0]; Int[n_] := If[n < 4, 0, Binomial[n, 4] + del[2, n] (-5n^3 + 45n^2 - 70n + 24)/24 - del[4, n] (3n/2) + del[6, n] (-45n^2 + 262n)/6 + del[12, n]*42n + del[18, n]*60n + del[24, n]*35n - del[30, n]*38n - del[42, n]*82n - del[60, n]*330n - del[84, n]*144n - del[90, n]*96n - del[120, n]*144n - del[210, n]*96n]; Select[ Range@ 759, PrimeQ@ Int@# &]

def is_A180686(k):
    return Integer(binomial(k,4) + (-5*k^3+45*k^2-70*k+24)*(k%2==0)/24
        - 3*k*(k%4==0)/2 + (-45*k^2+262*k)*(k%6==0)/6 + 42*k*(k%12==0)
        + 60*k*(k%18==0) + 35*k*(k%24==0) - 38*k*(k%30==0)
        - 82*k*(k%42==0) - 330*k*(k%60==0) - 144*k*(k%84==0)
        - 96*k*(k%90==0) - 144*k*(k%120==0) - 96*k*(k%210==0)).is_prime()
print([k for k in range(1, 1000) if is_A180686(k)])  # Robin Visser, Jul 29 2024

Name edited by Robin Visser, Jul 29 2024

A238822 Number of equilateral triangles bounded by the sides and diagonals of a regular 3n-gon.

Original entry on oeis.org

1, 14, 69, 188, 465, 882, 1645, 2648, 4257, 6170, 9141, 12564, 17329, 22778, 30045, 38192, 48705, 60318, 74917, 90740, 110481, 131714, 157389, 184968, 217825, 252902, 294165, 337988, 388977, 442710, 505021, 570464, 645249, 723758, 812805, 906012, 1011025, 1120658, 1243437, 1371080

Offset: 1

N. J. A. Sloane, Mar 07 2014

Andrew Howroyd, Table of n, a(n) for n = 1..100
Antreas Hatzipolakis and Giovanni Resta, Number of Equilateral Triangles in Regular 3n-gon, Postings to the Sequence Fans Mailing List, Apr 08 2013
Andrew Howroyd, PARI Program, Sep 2024.
Giovanni Resta, Illustration for a(3) = 69

Cf. A006561.

Conjecture: For odd n, a(n) = n*(n+1)^3/2 - 3*n^2. - Vilim Lendvaj, Jul 24 2024

a(6)-a(24) from Luca Petrone, Mar 28 2017
Corrected and extended by Vilim Lendvaj, Jul 24 2024
a(26) onwards from Andrew Howroyd, Sep 26 2024

cp's OEIS Frontend

A187781 Number of noncongruent polygonal regions in a regular n-gon with all diagonals drawn.

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A230281 The least possible number of intersection points of the diagonals in the interior of a convex n-gon with all diagonals drawn.

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A363174 Array read by rows: T(n,k) is the number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from k distinct vertices, with n >= 3, 3 <= k <= 6.

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A211382 Number of intersections of diagonals in the exterior of a regular n-gon.

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A211383 Number of intersections of diagonals in the interior and exterior of a regular n-gon.

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A351924 The number of vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

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A363173 Number of triangles inside a regular n-gon formed by intersecting line segments, considering all configurations of 3 line segments from 6 distinct vertices.

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Mathematica

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A108053 Maximum number of diagonals of a regular n-gon that meet at a non-center point.

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A180686 Positive integers k such that the number of intersections of diagonals in the interior of a regular k-gon is prime.

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