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

3,3

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.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Row sums of A124287 (k-gon triangle).

Cf. A293820 (polygons modulo rotations only).

Sequence in context: A062773 A319130 A329467 * A037246 A310019 A261998

Adjacent sequences: A293815 A293816 A293817 * A293819 A293820 A293821

Keyword

nonn

Author

James East, Oct 16 2017

Status

approved