A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.

1, 1, 3, 6, 13, 21, 37, 51, 84, 108, 166, 203, 294, 350, 486, 566, 759, 867, 1133, 1276, 1631, 1815, 2275, 2509, 3094, 3386, 4116, 4473, 5372, 5804, 6896, 7412, 8721, 9333, 10887, 11606, 13433, 14269, 16401, 17367, 19836, 20944, 23782, 25047, 28290, 29726, 33410, 35030, 39195, 41015

Offset

5,3

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=5..54.

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

Formula

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

Example

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.

Mathematica

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 60}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A293819 *)

Crossrefs

Column k=5 of A293819.

Cf. A293820 (polygons), A293821 (quadrilaterals), A293823 (hexagons).

Sequence in context: A064621 A330588 A194279 * A371917 A180750 A047172

Adjacent sequences: A293819 A293820 A293821 * A293823 A293824 A293825

Keyword

nonn

Author

James East, Oct 16 2017

Status

approved