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Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.
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%I #14 Oct 25 2020 14:16:48

%S 1,6,10,18,31,30,58,60,73,90,118,72,160,168,187,204,262,240,325,306,

%T 358,396,457,324,535,546,580,594,709,666,808,780,859,918,1012,780,

%U 1126,1140,1189,1212,1372,1308,1507,1458,1576,1656,1783,1464,1933,1950,2014,2034

%N Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.

%C The first ray from each point goes opposite to the direction to the center of the circle. Should a ray hit another point it is terminated there.

%C To produce the illustrations below, all pairwise intersections between the rays are calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit.

%H Lars Blomberg, <a href="/A338122/b338122.txt">Table of n, a(n) for n = 1..800</a>

%H Lars Blomberg, <a href="/A338122/a338122.png">Illustration for n=3</a>

%H Lars Blomberg, <a href="/A338122/a338122_1.png">Illustration for n=9</a>

%H Lars Blomberg, <a href="/A338122/a338122_2.png">Illustration for n=15</a>

%H Lars Blomberg, <a href="/A338122/a338122_3.png">Illustration for n=24</a>

%H Lars Blomberg, <a href="/A338122/a338122_4.png">Illustration for n=27</a>

%H Lars Blomberg, <a href="/A338122/a338122_5.png">Illustration for n=30</a>

%H Lars Blomberg, <a href="/A338122/a338122_6.png">Illustration for n=45</a>

%F a(n) = 2160-a(n-4)+a(n-12)+a(n-16)+a(n-60)+a(n-64)-a(n-72)-a(n-76), n>78. (conjectured)

%F From _Lars Blomberg_, Oct 25 2020: (Start)

%F Conjectured for 1 <= n <= 800.

%F Select the row in the table below for which r = n mod m. Then a(n)=(a*n^2 + b*n + c)/d.

%F +===========================================+

%F | r | m | a | b | c | d |

%F +-------------------------------------------+

%F | 1 | 12 | 3 | 11 | -10 | 4 |

%F | 2, 10 | 12 | 3 | 6 | | 4 |

%F | 3 | 12 | 3 | 5 | -2 | 4 |

%F | 5 | 12 | 3 | 11 | -6 | 4 |

%F | 6 | 12 | 3 | -2 | 24 | 4 |

%F | 7 | 12 | 3 | 11 | 8 | 4 |

%F | 9 | 12 | 3 | 5 | 4 | 4 |

%F | 11 | 12 | 3 | 11 | -12 | 4 |

%F | 4, 20 | 24 | 3 | | 24 | 4 |

%F | 8, 16 | 24 | 3 | | 48 | 4 |

%F | 0 | 120 | 3 | -26 | | 4 |

%F | 12, 36, 84, 108 | 120 | 3 | -26 | 168 | 4 |

%F | 24, 48, 72, 96 | 120 | 3 | -26 | 192 | 4 |

%F | 60 | 120 | 3 | -26 | -24 | 4 |

%F +===========================================+ (End)

%e For n=1 there are three rays that do not intersect, so a(1)=1.

%o (PARI)

%o a(n)=if( \

%o n%12==1,(3*n^2 + 11*n - 10)/4, \

%o n%12==2||n%12==10,(3*n^2 + 6*n)/4, \

%o n%12==3,(3*n^2 + 5*n - 2)/4, \

%o n%12==5,(3*n^2 + 11*n - 6)/4, \

%o n%12==6,(3*n^2 - 2*n + 24)/4, \

%o n%12==7,(3*n^2 + 11*n + 8)/4, \

%o n%12==9,(3*n^2 + 5*n + 4)/4, \

%o n%12==11,(3*n^2 + 11*n - 12)/4, \

%o n%24==4||n%24==20,(3*n^2 + 24)/4, \

%o n%24==8||n%24==16,(3*n^2 + 48)/4, \

%o n%120==0,(3*n^2 - 26*n)/4, \

%o n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 26*n + 168)/4, \

%o n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 26*n + 192)/4, \

%o n%120==60,(3*n^2 - 26*n - 24)/4, \

%o -1);

%o vector(800, n, a(n))

%Y Cf. A338041 (two start points), A338123 (vertices), A338124 (edges).

%K nonn

%O 1,2

%A _Lars Blomberg_, Oct 11 2020