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A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections. 5
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 (list; graph; refs; listen; history; text; internal format)
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: A064290 A064621 A194279 * A180750 A047172 A034734

Adjacent sequences:  A293819 A293820 A293821 * A293823 A293824 A293825

KEYWORD

nonn

AUTHOR

James East, Oct 16 2017

STATUS

approved

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Last modified December 15 15:44 EST 2018. Contains 318150 sequences. (Running on oeis4.)