OFFSET
4,3
COMMENTS
Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2).
A formula is given in Section 6 of the East and Niles article.
LINKS
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
FORMULA
Conjectures from Colin Barker, Nov 01 2017: (Start)
G.f.: x^3*(1 - x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)).
a(n) = (1/96)*(-3*(-1 + (-1)^n + 4*i*(-i)^n - 4*i*i^n) + (7 - 15*(-1)^n)*n + 3*(-1 + (-1)^n)*n^2 + 2*n^3) where i=sqrt(-1).
(End)
EXAMPLE
For example, there are 4 rotation-classes of perimeter-7 quadrilaterals: 3211, 3121, 3112, 2221. Note that 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.
MATHEMATICA
T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 4];
CROSSREFS
KEYWORD
nonn
AUTHOR
James East, Oct 16 2017
STATUS
approved