A338123 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of vertices thus created. See Comments for details.

3, 4, 15, 19, 33, 31, 63, 55, 78, 82, 120, 67, 162, 154, 189, 175, 261, 217, 327, 259, 360, 370, 456, 283, 534, 514, 579, 523, 705, 619, 807, 703, 858, 874, 1008, 691, 1122, 1090, 1185, 1111, 1365, 1237, 1503, 1339, 1572, 1594, 1776, 1339, 1926, 1882, 2007, 1891

Offset

1,1

Comments

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.

See A338122 for illustrations.

Links

Lars Blomberg, Table of n, a(n) for n = 1..800

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 3 <= 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 |

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

| 5 | 6 | 3 | 10 | 7 | 4 |

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

| 2, 10 | 12 | 3 | | 28 | 4 |

| 3 | 12 | 3 | 4 | 21 | 4 |

| 6 | 12 | 3 | -10 | 76 | 4 |

| 7 | 12 | 3 | 10 | 35 | 4 |

| 9 | 12 | 3 | 4 | 33 | 4 |

| 4, 20 | 24 | 3 | -12 | 76 | 4 |

| 8, 16 | 24 | 3 | -12 | 124 | 4 |

| 0 | 120 | 3 | -40 | -20 | 4 |

| 12, 36, 84, 108 | 120 | 3 | -40 | 316 | 4 |

| 24, 48, 72, 96 | 120 | 3 | -40 | 364 | 4 |

| 60 | 120 | 3 | -40 | -68 | 4 |

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

Example

For n=1 there are three rays that do not intersect, so a(1)=3.

Prog

(PARI)
a(n)=if( \
n%6==5, (3*n^2 + 10*n + 7)/4, \
n%12==1, (3*n^2 + 10*n + 11)/4, \
n%12==2||n%12==10, (3*n^2 + 28)/4, \
n%12==3, (3*n^2 + 4*n + 21)/4, \
n%12==6, (3*n^2 - 10*n + 76)/4, \
n%12==7, (3*n^2 + 10*n + 35)/4, \
n%12==9, (3*n^2 + 4*n + 33)/4, \
n%24==4||n%24==20, (3*n^2 - 12*n + 76)/4, \
n%24==8||n%24==16, (3*n^2 - 12*n + 124)/4, \
n%120==0, (3*n^2 - 40*n - 20)/4, \
n%120==12||n%120==36||n%120==84||n%120==108, (3*n^2 - 40*n + 316)/4, \
n%120==24||n%120==48||n%120==72||n%120==96, (3*n^2 - 40*n + 364)/4, \
n%120==60, (3*n^2 - 40*n - 68)/4, \
-1);
vector(798, n, a(n+2))

Crossrefs

Cf. A338042 (two start points), A338122 (regions), A338124 (edges).

Sequence in context: A325186 A053359 A056742 * A041435 A136210 A041819

Adjacent sequences: A338120 A338121 A338122 * A338124 A338125 A338126

Keyword

nonn

Author

Lars Blomberg, Oct 11 2020

Status

approved