A124286 Number of integer-sided hexagons having perimeter n.

0, 0, 0, 0, 0, 1, 1, 4, 7, 15, 25, 46, 72, 113, 172, 248, 360, 491, 686, 896, 1217, 1536, 2031, 2504, 3236, 3905, 4955, 5880, 7336, 8586, 10556, 12208, 14823, 16964, 20364, 23106, 27456, 30906, 36399, 40692, 47532, 52816, 61237, 67672, 77941, 85701

Offset

1,8

Comments

Rotations and reversals are counted only once. Note that this is different from A069907, which counts hexagons whose sides are nondecreasing.

Links

Table of n, a(n) for n=1..46.

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

Formula

Empirical g.f.: x^6*(x^13 +3*x^12 +6*x^11 +6*x^10 +10*x^9 +9*x^8 +12*x^7 +10*x^6 +8*x^5 +5*x^4 +4*x^3 +2*x^2 +x +1) / ((x -1)^6*(x +1)^5*(x^2 -x +1)*(x^2 +1)^2*(x^2 +x +1)^2). - Colin Barker, Oct 27 2013

Example

The four hexagons having perimeter 8 are (1,1,1,1,2,2), (1,1,1,2,1,2), (1,1,2,1,1,2) and (1,1,1,1,1,3).

Mathematica

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==6 && #[[1]]<Total[Rest[ # ]]&]; cnt=0; Do[cnt=cnt+Length[ListNecklaces[6, s[[i]], Dihedral]], {i, Length[s]}]; cnt, {n, 50}]

Crossrefs

Cf. A057886 (quadrilaterals), A124285 (pentagons), A124287 (k-gons).

Sequence in context: A039669 A109622 A269967 * A235603 A295728 A027419

Adjacent sequences: A124283 A124284 A124285 * A124287 A124288 A124289

Keyword

nonn

Author

T. D. Noe, Oct 24 2006

Status

approved