A338421 Place four points evenly spaced on a circle, draw n evenly spaced rays from each of the points, a(n) is the number of regions thus created. See Comments for details.

1, 4, 21, 16, 57, 44, 93, 36, 149, 132, 217, 176, 301, 268, 385, 208, 489, 452, 605, 528, 737, 684, 869, 532, 1021, 964, 1185, 1072, 1365, 1292, 1545, 1112, 1745, 1668, 1957, 1808, 2185, 2092, 2413, 1844, 2661, 2564, 2921, 2736, 3197, 3084, 3473, 2696, 3769

Offset

1,2

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.

To produce the illustrations below, all pairwise intersections between the rays are calculated and the maximum distance to the center, incremented by 20%, is taken as radius of a circle. Then all intersections between the rays and the circle defines a polygon which is used as limit.

Links

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

Lars Blomberg, Illustration for n=3

Lars Blomberg, Illustration for n=7

Lars Blomberg, Illustration for n=8

Lars Blomberg, Illustration for n=16

Lars Blomberg, Illustration for n=22

Lars Blomberg, Illustration for n=26

Lars Blomberg, Illustration for n=27

Lars Blomberg, Illustration for n=38

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 | -4 | 4 | 2 |

| 1 | 8 | 3 | 7 | -8 | 2 |

| 3 | 8 | 3 | 7 | -6 | 2 |

| 4 | 8 | 3 | -8 | 16 | 2 |

| 5 | 8 | 3 | 7 | 4 | 2 |

| 7 | 8 | 3 | 7 | -10 | 2 |

| 0 | 48 | 3 | -31 | -32 | 2 |

| 8, 40 | 48 | 3 | -31 | 128 | 2 |

| 16, 32 | 48 | 3 | -31 | 144 | 2 |

| 24 | 48 | 3 | -31 | 80 | 2 |

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

Example

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

Prog

(PARI)
a(n)={if(
n==1, 1,
n==2, 4,
n%4==2, (3*n^2 - 4*n + 4)/2,
n%8==1, (3*n^2 + 7*n - 8)/2,
n%8==3, (3*n^2 + 7*n - 6)/2,
n%8==4, (3*n^2 - 8*n + 16)/2,
n%8==5, (3*n^2 + 7*n + 4)/2,
n%8==7, (3*n^2 + 7*n - 10)/2,
n%48==0, (3*n^2 - 31*n - 32)/2,
n%48==8||n%48==40, (3*n^2 - 31*n + 128)/2,
n%48==16||n%48==32, (3*n^2 - 31*n + 144)/2,
n%48==24, (3*n^2 - 31*n + 80)/2,
-1); }
vector(642, n, a(n))

Crossrefs

Cf. A338122, A338422 (vertices), A338423 (edges).

Sequence in context: A333433 A202450 A144292 * A365101 A329404 A370979

Adjacent sequences: A338418 A338419 A338420 * A338422 A338423 A338424

Keyword

nonn

Author

Lars Blomberg, Oct 26 2020

Status

approved