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A293820 Number of integer-sided polygons having perimeter n, modulo rotations but not reflections.

%I A293820 #29 Jun 14 2018 04:00:56
%S A293820 1,1,3,5,11,19,43,75,155,287,567,1053,2063,3859,7455,14089,27083,
%T A293820 51463,98855,188697,362675,695155,1338087,2573235,4962875,9571195,
%U A293820 18496407,35759799,69240899,134154259,260235639,505163055,981575759,1908619755,3714304167,7233118641,14095779055
%N A293820 Number of integer-sided polygons having perimeter n, modulo rotations but not reflections.
%C A293820 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). These are row sums of A293819.
%C A293820 A formula is given in Section 6 of the East and Niles article.
%C A293820 The same article shows that a(n) is asymptotic to 2^n / n.
%H A293820 Andrew Howroyd, <a href="/A293820/b293820.txt">Table of n, a(n) for n = 3..200</a>
%H A293820 James East, Ron Niles, <a href="https://arxiv.org/abs/1710.11245">Integer polygons of given perimeter</a>, arXiv:1710.11245 [math.CO], 2017.
%F A293820 a(n) = (Sum_{d|n} phi(n/d)*2^d)/n - 1 - 2^floor(n/2). - _Andrew Howroyd_, Nov 21 2017
%e A293820 There are 11 polygons having perimeter 7: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111).
%t A293820 T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
%t A293820 a[n_] := Sum[T[n, k], {k, 3, n}]
%t A293820 Table[a[n], {n, 3, 40}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ *)
%o A293820 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*2^d)/n - 1 - 2^floor(n/2); \\ _Andrew Howroyd_, Nov 21 2017
%Y A293820 Cf. A008742 (triangles), A293818 (reflections allowed), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).
%Y A293820 Row sums of A293819 (k-gon triangle).
%K A293820 nonn
%O A293820 3,3
%A A293820 _James East_, Oct 16 2017