

A344907


Number of polygon edges formed when every pair of vertices of a regular (2n1)gon are joined by an infinite line.


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
See A344857 for other examples and images of the polygons.


LINKS



FORMULA

a(n) = 4*n^4  22*n^3 + 44*n^2  35*n + 9 (see Sidorenko link in A344857 for proof).
G.f.: 3*x^2*(1 + 5*x + 23*x^2 + 3*x^3)/(1  x)^5.
a(n) = 5*a(n1)  10*a(n2) + 10*a(n3)  5*a(n4) + a(n5) 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.


PROG

(Python)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



