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
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.
PROG
(Python)
def A344907(n): return n*(n*(n*(4*n - 22) + 44) - 35) + 9 # Chai Wah Wu, Sep 12 2021
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Scott R. Shannon, Jun 02 2021
STATUS
approved