A338042 Draw n rays from each of two distinct points in the plane; a(n) is the number of vertices thus created. See Comments for details.

2, 2, 4, 2, 8, 4, 14, 8, 22, 14, 32, 22, 44, 32, 58, 44, 74, 58, 92, 74, 112, 92, 134, 112, 158, 134, 184, 158, 212, 184, 242, 212, 274, 242, 308, 274, 344, 308, 382, 344, 422, 382, 464, 422, 508, 464, 554, 508, 602, 554, 652, 602, 704, 652, 758, 704, 814, 758

Offset

1,1

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.

See A338041 for illustrations.

The conjectures are true (see Fried link). - Sela Fried, Sep 21 2025

Links

Table of n, a(n) for n=1..58.

Sela Fried, On a polygon obtained by the intersection of two fans of rays, 2025.

Sela Fried, Proofs of Ten Conjectures From the OEIS, Journal of Integer Sequences, Vol. 29 (2026), Article 26.1.8. See pp. 18-22.

Formula

a(n) = (n^2 + 7)/4, n odd; (n^2 - 6*n + 16)/4, n even (conjectured).

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

G.f.: 2*x*(1 - x^2 - x^3 + 2*x^4)/((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)

Apparently a(n) = 2*(A008795(n-3) + 1). - Hugo Pfoertner, Oct 08 2020

E.g.f.: ((16 + x + x^2)*cosh(x) + (7 - 5*x + x^2)*sinh(x) - 16)/4. - Stefano Spezia, Apr 08 2026

Example

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

Prog

(PARI) a(n)=if(n%2==1, (n^2 + 7)/4, (n^2 - 6*n + 16)/4)
vector(200, n, a(n))

Crossrefs

Cf. A338041 (regions), A338043 (edges), A008795.

Sequence in context: A297112 A259192 A307313 * A131999 A113416 A303140

Adjacent sequences: A338039 A338040 A338041 * A338043 A338044 A338045

Keyword

nonn

Author

Lars Blomberg, Oct 08 2020

Status

approved