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

1,2

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.

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

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)

Example

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

Prog

(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))

Crossrefs

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

Sequence in context: A032641 A293555 A077626 * A169873 A363788 A079471

Adjacent sequences: A338119 A338120 A338121 * A338123 A338124 A338125

Keyword

nonn

Author

Lars Blomberg, Oct 11 2020

Status

approved