A338422 Place four points evenly spaced on 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.

4, 5, 24, 21, 64, 45, 96, 37, 152, 129, 216, 173, 304, 261, 384, 185, 488, 441, 600, 517, 736, 669, 864, 453, 1016, 945, 1176, 1053, 1360, 1269, 1536, 1025, 1736, 1641, 1944, 1781, 2176, 2061, 2400, 1717, 2648, 2529, 2904, 2701, 3184, 3045, 3456, 2465, 3752

Offset: 1

Author

Lars Blomberg, Oct 26 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 A338421 for illustrations.

Examples

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

Links

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

Crossrefs

Cf. A338123, A338421 (regions), A338423 (edges).

Programs

PARI

a(n)={ if(
n==1, 4,
n==2, 5,
n%4==2, (3*n^2 - 6*n + 18)/2,
n%4==3, (3*n^2 + 6*n + 3)/2,
n%8==1, (3*n^2 + 6*n + 7)/2,
n%8==4, (3*n^2 - 10*n + 34)/2,
n%8==5, (3*n^2 + 6*n + 23)/2,
n%48==0, (3*n^2 - 39*n - 110)/2,
n%48==8||n%48==40, (3*n^2 - 39*n + 194)/2,
n%48==16||n%48==32, (3*n^2 - 39*n + 226)/2,
n%48==24, (3*n^2 - 39*n + 114)/2,
-1); }
vector(642, n, a(n))

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 | -6 | 18 | 2 |

| 3 | 4 | 3 | 6 | 3 | 2 |

| 1 | 8 | 3 | 6 | 7 | 2 |

| 4 | 8 | 3 | -10 | 34 | 2 |

| 5 | 8 | 3 | 6 | 23 | 2 |

| 0 | 48 | 3 | -39 | -110 | 2 |

| 8, 40 | 48 | 3 | -39 | 194 | 2 |

| 16, 32 | 48 | 3 | -39 | 226 | 2 |

| 24 | 48 | 3 | -39 | 114 | 2 |

+==================================+