A338124 Place three points evenly spaced around a circle, draw n evenly spaced rays from each of the points, a(n) is the number of edges thus created. See Comments for details.

3, 9, 24, 36, 63, 60, 120, 114, 150, 171, 237, 138, 321, 321, 375, 378, 522, 456, 651, 564, 717, 765, 912, 606, 1068, 1059, 1158, 1116, 1413, 1284, 1614, 1482, 1716, 1791, 2019, 1470, 2247, 2229, 2373, 2322, 2736, 2544, 3009, 2796, 3147, 3249, 3558, 2802, 3858

Offset: 1

Author

Lars Blomberg, Oct 11 2020

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A338043 (two start points), A338122 (regions), A338123 (vertices).

Programs

PARI

a(n)=if( \
n%12==1||n%12==5,(6*n^2 + 21*n - 3)/4, \
n%12==2||n%12==10,(3*n^2 + 3*n + 12)/2, \
n%12==3,(6*n^2 + 9*n + 15)/4, \
n%12==6,(3*n^2 - 6*n + 48)/2, \
n%12==7,(6*n^2 + 21*n + 39)/4, \
n%12==9,(6*n^2 + 9*n + 33)/4, \
n%12==11,(6*n^2 + 21*n - 9)/4, \
n%24==4||n%24==20,(3*n^2 - 6*n + 48)/2, \
n%24==8||n%24==16,(3*n^2 - 6*n + 84)/2, \
n%120==0,(3*n^2 - 33*n - 12)/2, \
n%120==12||n%120==36||n%120==84||n%120==108,(3*n^2 - 33*n + 240)/2, \
n%120==24||n%120==48||n%120==72||n%120==96,(3*n^2 - 33*n + 276)/2, \
n%120==60,(3*n^2 - 33*n - 48)/2, \
-1);
vector(798, n, a(n+2))

Formula

a(n) = 4320-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 |

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

| 1, 5 | 12 | 6 | 21 | -3 | 4 |

| 2, 10 | 12 | 3 | 3 | 12 | 2 |

| 3 | 12 | 6 | 9 | 15 | 4 |

| 6 | 12 | 3 | -6 | 48 | 2 |

| 7 | 12 | 6 | 21 | 39 | 4 |

| 9 | 12 | 6 | 9 | 33 | 4 |

| 11 | 12 | 6 | 21 | -9 | 4 |

| 4, 20 | 24 | 3 | -6 | 48 | 2 |

| 8, 16 | 24 | 3 | -6 | 84 | 2 |

| 0 | 120 | 3 | -33 | -12 | 2 |

| 12, 36, 84, 108 | 120 | 3 | -33 | 240 | 2 |

| 24, 48, 72, 96 | 120 | 3 | -33 | 276 | 2 |

| 60 | 120 | 3 | -33 | -48 | 2 |

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