A338043 Draw n rays from each of two distinct points in the plane; a(n) is the number of edges thus created. See Comments for details.

2, 3, 10, 7, 22, 15, 38, 27, 58, 43, 82, 63, 110, 87, 142, 115, 178, 147, 218, 183, 262, 223, 310, 267, 362, 315, 418, 367, 478, 423, 542, 483, 610, 547, 682, 615, 758, 687, 838, 763, 922, 843, 1010, 927, 1102, 1015, 1198, 1107, 1298, 1203, 1402, 1303, 1510, 1407

Offset

1,1

Comments

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

See A338041 for illustrations.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Sela Fried, Proofs of Ten Conjectures from the OEIS, J. Int. Seq. 29 (2026), Article 26.1.8. See pp. 18 (Sect. 2.10), 21 (Cor. 19).

Formula

a(n) = (n^2 + 4*n - 1)/2, n odd; (n^2 - 2*n + 6)/2, n even (conjectured).

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

G.f.: x*(2 + x + 3*x^2 - 5*x^3 + 3*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: It appears that even terms are 4*A000096+2, odd terms 4*A000124-1.

Example

For n=1:    <-----x     x----->   so a(1)=2.
For n=2:    <-----x<--->x----->   so a(2)=3.

Mathematica

Rest@ CoefficientList[Series[x*(2 + x + 3*x^2 - 5*x^3 + 3*x^4)/((1 - x)^3*(1 + x)^2), {x, 0, 54}], x] (* Michael De Vlieger, Apr 07 2026 *)

Prog

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

Crossrefs

Cf. A338041 (regions), A338042 (vertices).

Cf. also A000096, A000124.

Sequence in context: A128531 A123167 A333176 * A141670 A278561 A369991

Adjacent sequences: A338040 A338041 A338042 * A338044 A338045 A338046

Keyword

nonn

Author

Lars Blomberg, Oct 08 2020

Status

approved