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

Author

Lars Blomberg, Oct 08 2020

Keywords

• nonn

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.

Examples

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

Crossrefs

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

Programs

PARI

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

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)

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