cp's OEIS Frontend

A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.

%I A293821 #31 Jan 29 2019 09:24:58
%S A293821 1,1,2,4,6,10,12,20,23,35,38,56,60,84,88,120,125,165,170,220,226,286,
%T A293821 292,364,371,455,462,560,568,680,688,816,825,969,978,1140,1150,1330,
%U A293821 1340,1540,1551,1771,1782,2024,2036,2300,2312,2600,2613,2925,2938,3276,3290,3654,3668,4060
%N A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.
%C A293821 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 A293821 A formula is given in Section 6 of the East and Niles article.
%H A293821 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 A293821 Conjectures from _Colin Barker_, Nov 01 2017: (Start)
%F A293821 G.f.: x^3*(1 - x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)).
%F A293821 a(n) = (1/96)*(-3*(-1 + (-1)^n + 4*i*(-i)^n - 4*i*i^n) + (7 - 15*(-1)^n)*n + 3*(-1 + (-1)^n)*n^2 + 2*n^3) where i=sqrt(-1).
%F A293821 (End)
%e A293821 For example, there are 4 rotation-classes of perimeter-7 quadrilaterals: 3211, 3121, 3112, 2221. Note that 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.
%t A293821 T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
%t A293821 a[n_] := T[n, 4];
%t A293821 Table[a[n], {n, 4, 59}] (* _Jean-François Alcover_, Jan 29 2019, after _Andrew Howroyd_ in A293819 *)
%Y A293821 Column k=4 of A293819.
%Y A293821 Cf. A008742 (triangles), A293820 (polygons), A293822 (pentagons).
%K A293821 nonn
%O A293821 4,3
%A A293821 _James East_, Oct 16 2017