A338423 Place four points evenly spaced on 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.

4, 8, 44, 36, 120, 88, 188, 72, 300, 260, 432, 348, 604, 528, 768, 392, 976, 892, 1204, 1044, 1472, 1352, 1732, 984, 2036, 1908, 2360, 2124, 2724, 2560, 3080, 2136, 3480, 3308, 3900, 3588, 4360, 4152, 4812, 3560, 5308, 5092, 5824, 5436, 6380, 6128, 6928, 5160

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 A338421 for illustrations.

Links

Lars Blomberg, Table of n, a(n) for n = 1..642

Formula

Conjectured for 3 <= n <= 642.

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 |

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

| 2 | 4 | 3 | -5 | 10 | 1 |

| 1 | 8 | 6 | 13 | -3 | 2 |

| 3 | 8 | 6 | 13 | -5 | 2 |

| 4 | 8 | 3 | -9 | 24 | 1 |

| 5 | 8 | 6 | 13 | 25 | 2 |

| 7 | 8 | 6 | 13 | -9 | 2 |

| 0 | 48 | 3 | -35 | -72 | 1 |

| 8, 40 | 48 | 3 | -35 | 160 | 1 |

| 16, 32 | 48 | 3 | -35 | 184 | 1 |

| 24 | 48 | 3 | -35 | 96 | 1 |

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

Example

For n=1 there are four rays that do not intersect, so a(1)=4.

Prog

(PARI)
a(n)={if(
n==1, 4,
n==2, 8,
n%4==2, 3*n^2 - 5*n + 10,
n%8==1, (6*n^2 + 13*n - 3)/2,
n%8==3, (6*n^2 + 13*n - 5)/2,
n%8==4, 3*n^2 - 9*n + 24,
n%8==5, (6*n^2 + 13*n + 25)/2,
n%8==7, (6*n^2 + 13*n - 9)/2,
n%48==0, 3*n^2 - 35*n - 72,
n%48==8||n%48==40, 3*n^2 - 35*n + 160,
n%48==16||n%48==32, 3*n^2 - 35*n + 184,
n%48==24, 3*n^2 - 35*n + 96,
-1); }
vector(642, n, a(n))

Crossrefs

Cf. A338124, A338421 (regions), A338422 (vertices).

Sequence in context: A137052 A111411 A137054 * A163343 A284972 A045639

Adjacent sequences: A338420 A338421 A338422 * A338424 A338425 A338426

Keyword

nonn

Author

Lars Blomberg, Oct 26 2020

Status

approved