OFFSET
6,3
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
G.f.: x^6*(1 + x + 5*x^3 + 10*x^4 + 7*x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9) / ((1 - x)^6*(1 + x)^5*(1 - x + x^2)*(1 + x + x^2)^2) (conjectured). - Colin Barker, Nov 01 2017
EXAMPLE
For example, there are 10 rotation-classes of perimeter-9 hexagons: 411111, 321111, 312111, 311211, 311121, 311112, 222111, 221211, 221121, 212121. Note that 321111 and 311112 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, 6];
CROSSREFS
KEYWORD
nonn
AUTHOR
James East, Oct 16 2017
STATUS
approved