A338041 Draw n rays from each of two distinct points in the plane; a(n) is the number of regions thus created. See Comments for details.

1, 2, 7, 6, 15, 12, 25, 20, 37, 30, 51, 42, 67, 56, 85, 72, 105, 90, 127, 110, 151, 132, 177, 156, 205, 182, 235, 210, 267, 240, 301, 272, 337, 306, 375, 342, 415, 380, 457, 420, 501, 462, 547, 506, 595, 552, 645, 600, 697, 650, 751, 702, 807, 756, 865, 812, 925

Offset

1,2

Comments

The rays are evenly spaced around each point. The first ray of one point goes opposite to the direction to the other point. Should a ray hit the other point it terminates there, that is, it is converted to a line segment.

To produce the illustrations below, all pairwise intersections between the rays is 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

Table of n, a(n) for n=1..57.

Lars Blomberg, Illustration for n = 3

Lars Blomberg, Illustration for n = 6

Lars Blomberg, Illustration for n = 7

Lars Blomberg, Illustration for n = 18

Lars Blomberg, Illustration for n = 19

Lars Blomberg, Illustration for n = 33

Formula

a(n) = (n^2 + 8*n - 5)/4, n odd; (n^2 + 2*n)/4, n even (conjectured).

Conjectured by Stefano Spezia, Oct 08 2020 after Lars Blomberg: (Start)

G.f.: x*(1 + x + 3*x^2 - 3*x^3)/((1 - x)^3*(1 + x)^2).

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)

Example

For n=1:    <-----x     x----->   so a(1)=1.
For n=2:    <-----x<--->x----->   so a(2)=2.

Prog

(PARI) a(n)=if(n%2==1, (n^2 + 8*n - 5)/4, (n^2 + 2*n)/4);
vector(200, n, a(n))

Crossrefs

Cf. A338042 (vertices), A338043 (edges).

Sequence in context: A089417 A168205 A265495 * A365007 A082017 A072985

Adjacent sequences: A338038 A338039 A338040 * A338042 A338043 A338044

Keyword

nonn

Author

Lars Blomberg, Oct 08 2020

Status

approved