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

1,1

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) = 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)

Example

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

Prog

(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))

Crossrefs

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

Sequence in context: A029488 A198681 A254010 * A024314 A120012 A352640

Adjacent sequences: A338121 A338122 A338123 * A338125 A338126 A338127

Keyword

nonn

Author

Lars Blomberg, Oct 11 2020

Status

approved