A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

0, 1, 16, 99, 352, 925, 2016, 3871, 6784, 11097, 17200, 25531, 36576, 50869, 68992, 91575, 119296, 152881, 193104, 240787, 296800, 362061, 437536, 524239, 623232, 735625, 862576, 1005291, 1165024, 1343077, 1540800, 1759591, 2000896, 2266209, 2557072, 2875075, 3221856, 3599101, 4008544, 4451967

Offset

1,3

Comments

This is the odd-indexed subsequence of A344857. See A344857 for images of the polygons.

Links

Table of n, a(n) for n=1..40.

John Elias, Illustration of initial terms: Cubic Vertex Fractal

J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.

Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = 2*n^4 - 11*n^3 + 23*n^2 - 21*n + 7.

G.f.: x^2*(1 + 11*x + 29*x^2 + 7*x^3)/(1 - x)^5. - Stefano Spezia, Jun 04 2021

Example

a(3) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total.

Prog

(Python)
def A344866(n): return n*(n*(n*(2*n - 11) + 23) - 21) + 7 # Chai Wah Wu, Sep 12 2021

Crossrefs

Cf. A344857 (number for even and odd n), A344311, A344938, A007678, A341735 (number inside the n-gon), A344899 (number of edges).

See also A347320.

Sequence in context: A214612 A283545 A297684 * A322722 A014762 A045784

Adjacent sequences: A344863 A344864 A344865 * A344867 A344868 A344869

Keyword

nonn,easy

Author

Scott R. Shannon, Jun 01 2021

Extensions

Edited by N. J. A. Sloane, Sep 12 2021

Status

approved