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%I #26 Jun 14 2018 04:11:47
%S 1,1,3,6,13,21,37,51,84,108,166,203,294,350,486,566,759,867,1133,1276,
%T 1631,1815,2275,2509,3094,3386,4116,4473,5372,5804,6896,7412,8721,
%U 9333,10887,11606,13433,14269,16401,17367,19836,20944,23782,25047,28290,29726,33410,35030,39195,41015
%N Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.
%C 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).
%C A formula is given in Section 6 of the East and Niles article.
%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.
%F G.f.: x^5*(1 + x - x^2 + 2*x^3 + 7*x^4 + 2*x^5 - 2*x^6 + x^8) / ((1 - x)^5*(1 + x)^4*(1 + x + x^2 + x^3 + x^4)) (conjectured). - _Colin Barker_, Nov 01 2017
%e For example, there are 6 rotation-classes of perimeter-8 pentagons: 32111, 31211, 31121, 31112, 22211, 22121. Note that 32111 and 31112 are reflections of each other, but these are not rotationally equivalent.
%t T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
%t a[n_] := T[n, 5];
%t Table[a[n], {n, 5, 60}] (* _Jean-François Alcover_, Jun 14 2018, after _Andrew Howroyd_ and A293819 *)
%Y Column k=5 of A293819.
%Y Cf. A293820 (polygons), A293821 (quadrilaterals), A293823 (hexagons).
%K nonn
%O 5,3
%A _James East_, Oct 16 2017
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