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Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.
4

%I #31 Jan 29 2019 09:24:58

%S 1,1,2,4,6,10,12,20,23,35,38,56,60,84,88,120,125,165,170,220,226,286,

%T 292,364,371,455,462,560,568,680,688,816,825,969,978,1140,1150,1330,

%U 1340,1540,1551,1771,1782,2024,2036,2300,2312,2600,2613,2925,2938,3276,3290,3654,3668,4060

%N Number of integer-sided quadrilaterals 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 Conjectures from _Colin Barker_, Nov 01 2017: (Start)

%F G.f.: x^3*(1 - x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)).

%F 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 (End)

%e 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 T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];

%t a[n_] := T[n, 4];

%t Table[a[n], {n, 4, 59}] (* _Jean-François Alcover_, Jan 29 2019, after _Andrew Howroyd_ in A293819 *)

%Y Column k=4 of A293819.

%Y Cf. A008742 (triangles), A293820 (polygons), A293822 (pentagons).

%K nonn

%O 4,3

%A _James East_, Oct 16 2017