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.

%I A006533 M1118 #75 Dec 04 2024 11:26:48
%S A006533 1,2,4,8,16,30,57,88,163,230,386,456,794,966,1471,1712,2517,2484,4048,
%T A006533 4520,6196,6842,9109,9048,12951,14014,17902,19208,24158,21510,31931,
%U A006533 33888,41449,43826,52956,52992,66712,70034,82993,86840,102091,97776,124314,129448,149986,155894,179447,179280
%N 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.
%C A006533 This sequence and A007678 are two equivalent ways of presenting the same sequence. - _N. J. A. Sloane_, Jan 23 2020
%C A006533 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
%D A006533 Jean Meeus, Wiskunde Post (Belgium), Vol. 10, 1972, pp. 62-63.
%D A006533 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006533 T. D. Noe, <a href="/A006533/b006533.txt">Table of n, a(n) for n = 1..1000</a>
%H A006533 M. F. Hasler, <a href="/A006561/a006561.html">Interactive illustration of A006561(n) & A006533(n)</a>; <a href="/A006533/a006533.png">colored version for n=6</a> <a href="/A006533/a006533_1.png">and for n=8</a>.
%H A006533 Sascha Kurz, <a href="http://www.mathe2.uni-bayreuth.de/sascha/oeis/drawing/drawing.html">m-gons in regular n-gons</a>
%H A006533 J. Meeus & N. J. A. Sloane, <a href="/A006532/a006532_1.pdf">Correspondence, 1974-1975</a>
%H A006533 Burkard Polster (Mathologer), <a href="https://www.youtube.com/watch?v=iJ8pnCO0nTY">The hardest "What comes next?" (Euler's pentagonal formula)</a>, Youtube video, Oct 17 2020.
%H A006533 Bjorn Poonen and Michael Rubinstein, <a href="https://doi.org/10.1137/S0895480195281246">The Number of Intersection Points Made by the Diagonals of a Regular Polygon</a>, SIAM J. Discrete Mathematics, Vol. 11, No. 1 (1998), pp. 135-156. <a href="http://math.mit.edu/~poonen/papers/ngon.pdf">(Author's copy.)</a>.
%H A006533 Bjorn Poonen and Michael Rubinstein, <a href="https://arxiv.org/abs/math/9508209">The number of intersection points made by the diagonals of a regular polygon</a>, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
%H A006533 Bjorn Poonen and Michael Rubinstein, <a href="http://math.mit.edu/~poonen/papers/ngon.m">Mathematica programs for these sequences</a>
%H A006533 Michael Rubinstein, <a href="/A006561/a006561_3.pdf">Drawings for n=4,5,6,...</a>
%H A006533 Prasad Balakrishnan Warrier, <a href="https://indianmathsoc.org/pdf/Mathstudent-V93-part3-4-2024.pdf#page=34">The physiognomy of the Erdős-Szekeres conjecture (happy ending problem)</a>, Math. Student (Indian Math. Soc., 2024) Vol. 93, Nos. 3-4, 28-48.
%H A006533 <a href="/index/Pol#Poonen">Sequences formed by drawing all diagonals in regular polygon</a>
%F A006533 Poonen and Rubinstein give an explicit formula for a(n) (see Mma code).
%F A006533 a(n) = A007678(n) + n. - _T. D. Noe_, Dec 23 2006
%t A006533 del[m_,n_]:=If[Mod[n,m]==0,1,0];
%t A006533 R[n_]:=(n^4-6n^3+23n^2-18n+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;
%t A006533 Table[R[n], {n,1,1000}] (* _T. D. Noe_, Dec 21 2006 *)
%o A006533 (Python)
%o A006533 def d(n,m): return not n % m
%o A006533 def A006533(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 - 36*n)//48 + 1 # _Chai Wah Wu_, Mar 08 2021
%o A006533 (PARI) apply( {A006533(n)=if(n%2, (((n-6)*n+23)*n-18)*n/24+1, ((n^3/2 -17*n^2/4 +22*n -if(n%4, 19, 28) +!(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
%Y A006533 Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
%K A006533 nonn,easy,nice
%O A006533 1,2
%A A006533 _N. J. A. Sloane_, Bjorn Poonen (poonen(AT)math.princeton.edu)
%E A006533 Added more terms from b-file. - _N. J. A. Sloane_, Jan 23 2020
%E A006533 Edited definition. - _N. J. A. Sloane_, Mar 17 2024