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Number of integer-sided polygons having perimeter n, modulo rotations and reflections.
3

%I #27 Jun 14 2018 05:21:19

%S 1,1,3,5,10,16,32,54,102,180,336,607,1144,2098,3960,7397,14022,26452,

%T 50404,95821,183322,350554,673044,1292634,2489502,4797694,9264396,

%U 17904220,34652962,67125898,130182972,252679320,490918440,954505718,1857413460,3616951513,7048412792,13744169104

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

%C 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.

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

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

%H Andrew Howroyd, <a href="/A293818/b293818.txt">Table of n, a(n) for n = 3..200</a>

%H James East, Ron Niles, <a href="https://arxiv.org/abs/1710.11245">Integer polygons of given perimeter</a>, arXiv:1710.11245 [math.CO], 2017.

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

%t 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] ];

%t Table[a[n], {n, 3, 40}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ *)

%o (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

%Y Row sums of A124287 (k-gon triangle).

%Y Cf. A293820 (polygons modulo rotations only).

%K nonn

%O 3,3

%A _James East_, Oct 16 2017