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
Examples
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.
Links
- Harvey P. Dale, Table of n, a(n) for n = 1..1000
- 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).
Crossrefs
Programs
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Mathematica
LinearRecurrence[{5,-10,10,-5,1},{0,1,16,99,352},40] (* Harvey P. Dale, Jun 01 2025 *) -
Python
def A344866(n): return n*(n*(n*(2*n - 11) + 23) - 21) + 7 # Chai Wah Wu, Sep 12 2021
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
Extensions
Edited by N. J. A. Sloane, Sep 12 2021
Comments