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

1,1

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 | -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 |

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

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, 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))

Crossrefs

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

Sequence in context: A341586 A010302 A361981 * A171885 A331261 A063986

Adjacent sequences: A338419 A338420 A338421 * A338423 A338424 A338425

Keyword

nonn

Author

Lars Blomberg, Oct 26 2020

Status

approved