A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

1, 1, 3, 5, 10, 16, 32, 54, 102, 180, 336, 607, 1144, 2098, 3960, 7397, 14022, 26452, 50404, 95821, 183322, 350554, 673044, 1292634, 2489502, 4797694, 9264396, 17904220, 34652962, 67125898, 130182972, 252679320, 490918440, 954505718, 1857413460, 3616951513, 7048412792, 13744169104

Offset: 3

Author

James East, Oct 16 2017

Keywords

• nonn

Comments

Rotations and reversals are counted only once. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. These are row sums of A124287.

A formula is proved in Theorem 1.6 of the East and Niles article.

The same article shows that a(n) is asymptotic to 2^(n-1) / n.

Examples

			There are 10 polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..200
• James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Crossrefs

Row sums of A124287 (k-gon triangle).

Cf. A293820 (polygons modulo rotations only).

Programs

Mathematica

a[n_] := DivisorSum[n, EulerPhi[n/#]*2^# &]/(2*n) + 2^Floor[(n - 3)/2] - If[Mod[n, 4] < 2, 3*2^Floor[(n - 4)/4], 2^Floor[(n + 2)/4] ];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

PARI

a(n)={sumdiv(n, d, eulerphi(n/d)*2^d)/(2*n) + 2^floor((n-3)/2) - if(n%4<2, 3*2^floor((n-4)/4), 2^floor((n+2)/4))} \\ Andrew Howroyd, Nov 21 2017