A006600 -id:A006600 - cp's OEIS frontend

A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

0, 1, 12, 30, 128, 147, 264, 1056, 600, 825, 2380, 1482, 1932, 9635, 3024, 3672, 8484, 5301, 6300, 19474, 8580, 9867, 20744, 12900, 14664, 30141, 18564, 20706, 62200, 25575, 28320, 54956, 34272, 37485, 62868, 44622, 48564, 86359, 57000, 61500, 117068, 71337

Offset: 2

Jonathan Sondow, Jul 25 2015

nonn

Same as (total number of triangles visible in convex 2n-gon with all diagonals drawn in general position) - (total number of triangles visible in regular 2n-gon with all diagonals drawn).
Number of triple-crossings of diagonals in the regular 2n+1-gon is 0.
See Sillke 1998 (where a(n) is called "T(2n)") for explanations and extensive annotated references.
See A005732 and A006600 for more comments, references, links, formulas, examples, programs, and lists from which to compute a(n) = A005732(2n) - A006600(2n) up to n = 500.

			With only 2 diagonals in a 4-gon, there can be no triple-crossings, so a(2) = 0.

T. Sillke, Number of triangles for a convex n-gon, 1998.

Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.

a(n) = A005732(2n) - A006600(2n).

A364828 a(n) is the number of regions inside a circle after inscribing (always starting from the same point), for each divisor d_i of n greater than 1, a regular d_i-gon.

Original entry on oeis.org

1, 2, 4, 6, 6, 12, 8, 14, 13, 18, 12, 36, 14, 24, 28, 30, 18, 46, 20, 52, 36, 36, 24, 88, 31, 42, 40, 66, 30, 114, 32, 62, 52, 54, 56, 140, 38, 60, 60, 128, 42, 144, 44, 94, 102, 72, 48, 196, 57, 104, 76, 108, 54, 152, 80, 162, 84, 90, 60, 360, 62, 96, 132, 126

Offset: 1

Paolo Xausa, Aug 09 2023

nonn

Inspired by an X (or Twitter) post by Matt Henderson (see links section).

Matt Henderson, X post, 2023.
Paolo Xausa, Illustration of first terms.
Paolo Xausa, Illustration of a(60).

Cf. A006600, A006533, A364829, A364830, A364838.

a(n) = n + 1, if n is prime.
a(n) = 2*(n-1), if n > 1 is a power of 2.
a(n) = A364829(n) + n.

A364829 a(n) is the number of regions inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

Original entry on oeis.org

0, 0, 1, 2, 1, 6, 1, 6, 4, 8, 1, 24, 1, 10, 13, 14, 1, 28, 1, 32, 15, 14, 1, 64, 6, 16, 13, 38, 1, 84, 1, 30, 19, 20, 21, 104, 1, 22, 21, 88, 1, 102, 1, 50, 57, 26, 1, 148, 8, 54, 25, 56, 1, 98, 25, 106, 27, 32, 1, 300, 1, 34, 69, 62, 27, 134, 1, 68, 31, 144

Offset: 1

Paolo Xausa, Aug 09 2023

nonn

Inspired by an X (or Twitter) post by Matt Henderson (see links section).

This is A364828 without the outer circle.

Matt Henderson, X post, 2023.
Paolo Xausa, Illustration of first terms.
Paolo Xausa, Illustration of a(60).

Cf. A006600, A007678, A364828, A364830, A364838.

a(n) = 1, if n is prime.
a(n) = n - 2, if n > 1 is a power of 2.
a(n) = A364828(n) - n.

A203016 Numbers congruent to {1, 2, 3, 4} mod 6, multiplied by 3.

Original entry on oeis.org

3, 6, 9, 12, 21, 24, 27, 30, 39, 42, 45, 48, 57, 60, 63, 66, 75, 78, 81, 84, 93, 96, 99, 102, 111, 114, 117, 120, 129, 132, 135, 138, 147, 150, 153, 156, 165, 168, 171, 174, 183, 186, 189, 192, 201, 204, 207, 210, 219, 222, 225, 228, 237, 240, 243, 246, 255, 258, 261, 264, 273, 276, 279, 282, 291, 294, 297

Offset: 1

N. J. A. Sloane, Dec 27 2011

Appears to coincide with the list of numbers n such that A006600(n) is not a multiple of n. Equals A047227 multiplied by 3.

Colin Foster, Peripheral mathematical knowledge, For the Learning of Mathematics, vol. 31, #3 (November, 2011), pp. 24-28.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A006600, A047227, A047235, A047241.

[3*n : n in [0..100] | n mod 6 in [1..4]]; // Wesley Ivan Hurt, Jun 07 2016

A203016:=n->3*(6*n-5-I^(2*n)+(1+I)*I^(1-n)+(1-I)*I^(n-1))/4: seq(A203016(n), n=1..100); # Wesley Ivan Hurt, Jun 07 2016

3 Select[Range[100], MemberQ[{1, 2, 3, 4}, Mod[#, 6]] &] (* Wesley Ivan Hurt, Jun 07 2016 *)

From Wesley Ivan Hurt, Jun 07 2016: (Start)
G.f.: 3*x*(1+x+x^2+x^3+2*x^4)/((x-1)^2*(1+x+x^2+x^3)).
a(n) = 3*(6*n-5-i^(2*n)+(1+i)*i^(1-n)+(1-i)*i^(n-1))/4 where i=sqrt(-1).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = 3*A047235(k), a(2k-1) = 3*A047241(k). (End)
E.g.f.: 3*(4 + sin(x) - cos(x) + (3*x - 2)*sinh(x) + 3*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 07 2016

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).

A036403 Number of equilateral triangles whose vertices (whether connected by lines or not) lie at intersection points resulting from drawing lines connecting every pair of vertices of a regular 3n-gon (and extending beyond the polygon).

Original entry on oeis.org

1, 126, 3927, 33156, 97115, 641916, 537607, 4222280, 1744695, 20962830, 4003241, 42626916

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

Given a regular 3n-gon, draw a line, extending beyond the polygon, through every pair of vertices; a(n) is the number of distinct equilateral triangles whose vertices lie at three of the resulting intersection points (whether the three points are connected by lines or not).

			Drawing lines connecting every pair of vertices on a regular hexagon (6-gon) and extending those lines beyond the polygon results in 37 distinct intersection points. Of the 37 * 36 * 35 / 3! = 7770 sets of 3 of those intersection points that could be selected, there are 126 sets of 3 intersection points such that, if the 3 points were connected by line segments, the resulting triangle would be equilateral, so a(2)=126.

Computed by Ilan Mayer (ilan(AT)isgtec.com).

Cf. A006600.

Added a(5) through a(8), corrected definition and comment and provided example, after receiving clarification Oct 22 2008 from Ilan Mayer (who had originally computed the sequence) regarding its definition. - Jon E. Schoenfield, Oct 23 2008

a(9)-a(12) from Jon E. Schoenfield, Oct 26 2008

A262248 Number of intersections of diagonals in the interior of a regular p-gon where p is the n-th prime.

Original entry on oeis.org

0, 0, 5, 35, 330, 715, 2380, 3876, 8855, 23751, 31465, 66045, 101270, 123410, 178365, 292825, 455126, 521855, 766480, 971635, 1088430, 1502501, 1837620, 2441626, 3464840, 4082925, 4421275, 5160610, 5563251, 6438740, 10334625, 11716640, 14043870

Offset: 1

Altug Alkan, Sep 16 2015

This is binomial(prime(n),4). - N. J. A. Sloane, May 17 2020
Subsequence of A006561.
a(n) = prime(n) only for n = 3.

			For prime(2)=3, there is no intersection of diagonals in the interior of a regular triangle, so a(2)=0.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A006561, A006600, A007569, A126995.

[(NthPrime(n)^4-6*(NthPrime(n)^3)+11*NthPrime(n)^2- 6*NthPrime(n))/24: n in [1..40]]; // Vincenzo Librandi, Sep 17 2015

Table[(Prime[n]^4 - 6 (Prime[n]^3) + 11 Prime[n]^2 - 6 Prime[n])/24, {n, 50}] (* Vincenzo Librandi, Sep 17 2015 *)
(#^4-6#^3+11#^2-6#)/24&/@Prime[Range[40]] (* Harvey P. Dale, Jun 17 2022 *)

a(n) = my(p=prime(n)); p*(p^3 - 6*p^2 + 11*p - 6)/24;
vector(40, n, a(n))

a(n) = (prime(n)^4 - 6*prime(n)^3 + 11*prime(n)^2 - 6*prime(n))/24.

a(n) = A006561(A000040(n)).

A176357 Partial sums of A036403.

Original entry on oeis.org

1, 127, 4054, 37210, 134325, 776241, 1313848, 5536128, 7280823, 28243653, 32246894, 74873810

Offset: 1

Jonathan Vos Post, Apr 15 2010

Partial sums of number of equilateral triangles whose vertices (whether connected by lines or not) lie at intersection points resulting from drawing lines connecting every pair of vertices of a regular 3n-gon (and extending beyond the polygon).

			a(4) = 1 + 126 + 3927 + 33156 = 37210 = 2 * 5 * 61^2.

Cf. A006600, A036403.

a(n) = Sum_{i=1..n} A036403(i).

A341441 Total number of triangles visible in a regular (2n+1)-gon with all diagonals drawn.

Original entry on oeis.org

1, 35, 287, 1302, 4257, 11297, 25935, 53516, 101745, 181279, 306383, 495650, 772785, 1167453, 1716191, 2463384, 3462305, 4776219, 6479551, 8659118, 11415425, 14864025, 19136943, 24384164, 30775185, 38500631, 47773935, 58833082, 71942417, 87394517, 105512127

Offset: 1

Edward Porcella, Feb 11 2021

For n=1, an equilateral triangle, there is no diagonal, and thus the polygon itself is the only triangle.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Bisection (odd part) of A005732 and of A006600.

a(n) = n*(2*n+1)*(2*n-1)*(2*n^3+21*n^2-2*n+9)/90.
G.f.: x*(x^5+20*x^4+7*x^3-63*x^2-28*x-1)/(x-1)^7. - Alois P. Heinz, Feb 11 2021
E.g.f.: exp(x)*x*(90 + 1485*x + 2775*x^2 + 1350*x^3 + 204*x^4 + 8*x^5)/90. - Stefano Spezia, Feb 12 2021

cp's OEIS Frontend

A260417 Number of triple-crossings of diagonals in the regular 2n-gon.

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A364828 a(n) is the number of regions inside a circle after inscribing (always starting from the same point), for each divisor d_i of n greater than 1, a regular d_i-gon.

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A364829 a(n) is the number of regions inside a regular n-gon after inscribing (always starting from the same point), for each proper divisor d_i of n greater than 1, a regular d_i-gon.

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A203016 Numbers congruent to {1, 2, 3, 4} mod 6, multiplied by 3.

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Mathematica

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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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A036403 Number of equilateral triangles whose vertices (whether connected by lines or not) lie at intersection points resulting from drawing lines connecting every pair of vertices of a regular 3n-gon (and extending beyond the polygon).

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A262248 Number of intersections of diagonals in the interior of a regular p-gon where p is the n-th prime.

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Magma

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A176357 Partial sums of A036403.

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A341441 Total number of triangles visible in a regular (2n+1)-gon with all diagonals drawn.