A338122 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.

1, 6, 10, 18, 31, 30, 58, 60, 73, 90, 118, 72, 160, 168, 187, 204, 262, 240, 325, 306, 358, 396, 457, 324, 535, 546, 580, 594, 709, 666, 808, 780, 859, 918, 1012, 780, 1126, 1140, 1189, 1212, 1372, 1308, 1507, 1458, 1576, 1656, 1783, 1464, 1933, 1950, 2014, 2034

Offset: 1

Author

Lars Blomberg, Oct 11 2020

Keywords

• nonn

Comments

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.

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.

Examples

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

Links

• Lars Blomberg, Table of n, a(n) for n = 1..800
• Lars Blomberg, Illustration for n=3
• Lars Blomberg, Illustration for n=9
• Lars Blomberg, Illustration for n=15
• Lars Blomberg, Illustration for n=24
• Lars Blomberg, Illustration for n=27
• Lars Blomberg, Illustration for n=30
• Lars Blomberg, Illustration for n=45

Crossrefs

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

Programs

PARI

a(n)=if( \
n%12==1,(3*n^2 + 11*n - 10)/4, \
n%12==2||n%12==10,(3*n^2 + 6*n)/4, \
n%12==3,(3*n^2 + 5*n - 2)/4, \
n%12==5,(3*n^2 + 11*n - 6)/4, \
n%12==6,(3*n^2 - 2*n + 24)/4, \
n%12==7,(3*n^2 + 11*n + 8)/4, \
n%12==9,(3*n^2 + 5*n + 4)/4, \
n%12==11,(3*n^2 + 11*n - 12)/4, \
n%24==4||n%24==20,(3*n^2 + 24)/4, \
n%24==8||n%24==16,(3*n^2 + 48)/4, \
n%120==0,(3*n^2 - 26*n)/4, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 26*n + 168)/4, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 26*n + 192)/4, \
n%120==60,(3*n^2 - 26*n - 24)/4, \
-1);
vector(800, n, a(n))

Formula

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)

From Lars Blomberg, Oct 25 2020: (Start)

Conjectured for 1 <= n <= 800.

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

+===========================================+

| r | m | a | b | c | d |

+-------------------------------------------+

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

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

| 3 | 12 | 3 | 5 | -2 | 4 |

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

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

| 7 | 12 | 3 | 11 | 8 | 4 |

| 9 | 12 | 3 | 5 | 4 | 4 |

| 11 | 12 | 3 | 11 | -12 | 4 |

| 4, 20 | 24 | 3 | | 24 | 4 |

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

| 0 | 120 | 3 | -26 | | 4 |

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

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

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

+===========================================+ (End)