A124285 Number of integer-sided pentagons having perimeter n.

0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 23, 29, 48, 60, 92, 109, 158, 186, 258, 296, 397, 451, 589, 658, 841, 933, 1169, 1283, 1582, 1728, 2100, 2275, 2732, 2948, 3502, 3756, 4419, 4725, 5511, 5866, 6789, 7207, 8283, 8761, 10006, 10560, 11990, 12617, 14250, 14968

Offset

1,7

Comments

Rotations and reversals are counted only once. Note that this is different from A069906, which counts pentagons whose sides are nondecreasing.

Links

Table of n, a(n) for n=1..50.

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

Formula

Empirical g.f.: -x^5*(x^12 +2*x^9 +2*x^8 +2*x^7 +5*x^6 +3*x^5 +2*x^4 +2*x^3 +x^2 +x +1) / ((x -1)^5*(x +1)^4*(x^2 +1)^2*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Oct 27 2013

Example

The three pentagons having perimeter 7 are (1,1,1,2,2), (1,1,2,1,2) and (1,1,1,1,3).

Mathematica

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==5 && #[[1]]<Total[Rest[ # ]]&]; cnt=0; Do[cnt=cnt+Length[ListNecklaces[5, s[[i]], Dihedral]], {i, Length[s]}]; cnt, {n, 50}]

Crossrefs

Cf. A057886 (quadrilaterals), A124286 (hexagons), A124287 (k-gons).

Sequence in context: A225288 A167930 A326981 * A131326 A089300 A250111

Adjacent sequences: A124282 A124283 A124284 * A124286 A124287 A124288

Keyword

nonn

Author

T. D. Noe, Oct 24 2006

Status

approved