cp's OEIS Frontend

A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

%I A293823 #30 Jan 29 2019 09:25:10
%S A293823 1,1,4,10,21,41,74,126,196,314,448,672,912,1302,1692,2334,2937,3927,
%T A293823 4828,6292,7579,9679,11466,14378,16808,20748,23968,29198,33388,40188,
%U A293823 45564,54264,61047,72033,80484,94164,104587,121429,134134,154672,170016,194810,213200,242880,264730,300002
%N A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.
%C A293823 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 A293823 A formula is given in Section 6 of the East and Niles article.
%H A293823 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 A293823 G.f.: x^6*(1 + x + 5*x^3 + 10*x^4 + 7*x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9) / ((1 - x)^6*(1 + x)^5*(1 - x + x^2)*(1 + x + x^2)^2) (conjectured). - _Colin Barker_, Nov 01 2017
%e A293823 For example, there are 10 rotation-classes of perimeter-9 hexagons: 411111, 321111, 312111, 311211, 311121, 311112, 222111, 221211, 221121, 212121. Note that 321111 and 311112 are reflections of each other, but these are not rotationally equivalent.
%t A293823 T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
%t A293823 a[n_] := T[n, 6];
%t A293823 Table[a[n], {n, 6, 51}] (* _Jean-François Alcover_, Jan 29 2019, after _Andrew Howroyd_ in A293819 *)
%Y A293823 Column k=6 of A293819.
%Y A293823 Cf. A293820 (polygons), A293822 (pentagons).
%K A293823 nonn
%O A293823 6,3
%A A293823 _James East_, Oct 16 2017