A338421 - OEIS

%I #16 Oct 29 2020 19:24:31

%S 1,4,21,16,57,44,93,36,149,132,217,176,301,268,385,208,489,452,605,

%T 528,737,684,869,532,1021,964,1185,1072,1365,1292,1545,1112,1745,1668,

%U 1957,1808,2185,2092,2413,1844,2661,2564,2921,2736,3197,3084,3473,2696,3769

%N Place four points evenly spaced on 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 rays are evenly spaced around each point. 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="/A338421/b338421.txt">Table of n, a(n) for n = 1..642</a>

%H Lars Blomberg, <a href="/A338421/a338421.png">Illustration for n=3</a>

%H Lars Blomberg, <a href="/A338421/a338421_1.png">Illustration for n=7</a>

%H Lars Blomberg, <a href="/A338421/a338421_2.png">Illustration for n=8</a>

%H Lars Blomberg, <a href="/A338421/a338421_3.png">Illustration for n=16</a>

%H Lars Blomberg, <a href="/A338421/a338421_4.png">Illustration for n=22</a>

%H Lars Blomberg, <a href="/A338421/a338421_5.png">Illustration for n=26</a>

%H Lars Blomberg, <a href="/A338421/a338421_6.png">Illustration for n=27</a>

%H Lars Blomberg, <a href="/A338421/a338421_7.png">Illustration for n=38</a>

%F Conjectured for 3 <= n <= 642.

%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 |      2 |  4 | 3 |  -4 |   4 | 2 |

%F |      1 |  8 | 3 |   7 |  -8 | 2 |

%F |      3 |  8 | 3 |   7 |  -6 | 2 |

%F |      4 |  8 | 3 |  -8 |  16 | 2 |

%F |      5 |  8 | 3 |   7 |   4 | 2 |

%F |      7 |  8 | 3 |   7 | -10 | 2 |

%F |      0 | 48 | 3 | -31 | -32 | 2 |

%F |  8, 40 | 48 | 3 | -31 | 128 | 2 |

%F | 16, 32 | 48 | 3 | -31 | 144 | 2 |

%F |     24 | 48 | 3 | -31 |  80 | 2 |

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

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

%o (PARI)

%o a(n)={if(

%o n==1,1,

%o n==2,4,

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

%o n%8==1,(3*n^2 + 7*n - 8)/2,

%o n%8==3,(3*n^2 + 7*n - 6)/2,

%o n%8==4,(3*n^2 - 8*n + 16)/2,

%o n%8==5,(3*n^2 + 7*n + 4)/2,

%o n%8==7,(3*n^2 + 7*n - 10)/2,

%o n%48==0,(3*n^2 - 31*n - 32)/2,

%o n%48==8||n%48==40,(3*n^2 - 31*n + 128)/2,

%o n%48==16||n%48==32,(3*n^2 - 31*n + 144)/2,

%o n%48==24,(3*n^2 - 31*n + 80)/2,

%o -1);}

%o vector(642, n, a(n))

%Y Cf. A338122, A338422 (vertices), A338423 (edges).

%K nonn

%O 1,2

%A _Lars Blomberg_, Oct 26 2020