A344907 - OEIS

A344907

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

0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575

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OFFSET

1,2

COMMENTS

This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.

A bisection of A344899. - N. J. A. Sloane, Sep 12 2021

See A344857 for other examples and images of the polygons.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

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

FORMULA

a(n) = 4*n^4 - 22*n^3 + 44*n^2 - 35*n + 9 (see Sidorenko link in A344857 for proof).

From Stefano Spezia, Jun 10 2021: (Start)

G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

EXAMPLE

a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed.

MATHEMATICA

A344907[n_] := (n-1)*(2*n-1)*(2*(n-4)*n + 9); Array[A344907, 50] (* Paolo Xausa, Jul 02 2026 *)

(* Alternative: *)

LinearRecurrence[{5, -10, 10, -5, 1}, {0, 3, 30, 189, 684}, 50] (* Paolo Xausa, Jul 02 2026 *)

PROG

(Python)

def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # Chai Wah Wu, Sep 12 2021

CROSSREFS

Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.