A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

1, 1, 4, 10, 21, 41, 74, 126, 196, 314, 448, 672, 912, 1302, 1692, 2334, 2937, 3927, 4828, 6292, 7579, 9679, 11466, 14378, 16808, 20748, 23968, 29198, 33388, 40188, 45564, 54264, 61047, 72033, 80484, 94164, 104587, 121429, 134134, 154672, 170016, 194810, 213200, 242880, 264730, 300002

Offset

6,3

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=6..51.

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

Formula

G.f.: x^6*(1 + x + 5*x^3 + 10*x^4 + 7*x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9) / ((1 - x)^6*(1 + x)^5*(1 - x + x^2)*(1 + x + x^2)^2) (conjectured). - Colin Barker, Nov 01 2017

Example

For example, there are 10 rotation-classes of perimeter-9 hexagons: 411111, 321111, 312111, 311211, 311121, 311112, 222111, 221211, 221121, 212121. Note that 321111 and 311112 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, 6];
Table[a[n], {n, 6, 51}] (* Jean-François Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

Crossrefs

Column k=6 of A293819.

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

Sequence in context: A266355 A265053 A266371 * A266354 A121497 A132925

Adjacent sequences: A293820 A293821 A293822 * A293824 A293825 A293826

Keyword

nonn

Author

James East, Oct 16 2017

Status

approved