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A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections. 4
1, 1, 2, 4, 6, 10, 12, 20, 23, 35, 38, 56, 60, 84, 88, 120, 125, 165, 170, 220, 226, 286, 292, 364, 371, 455, 462, 560, 568, 680, 688, 816, 825, 969, 978, 1140, 1150, 1330, 1340, 1540, 1551, 1771, 1782, 2024, 2036, 2300, 2312, 2600, 2613, 2925, 2938, 3276, 3290, 3654, 3668, 4060 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,3

COMMENTS

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).

A formula is given in Section 6 of the East and Niles article.

LINKS

Table of n, a(n) for n=4..59.

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

FORMULA

Conjectures from Colin Barker, Nov 01 2017: (Start)

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

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).

(End)

EXAMPLE

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.

MATHEMATICA

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

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

Table[a[n], {n, 4, 59}] (* Jean-Fran├žois Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

CROSSREFS

Column k=4 of A293819.

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

Sequence in context: A064374 A000885 A068336 * A194944 A133871 A068514

Adjacent sequences:  A293818 A293819 A293820 * A293822 A293823 A293824

KEYWORD

nonn

AUTHOR

James East, Oct 16 2017

STATUS

approved

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Last modified August 13 00:27 EDT 2020. Contains 336441 sequences. (Running on oeis4.)