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A124286
Number of integer-sided hexagons having perimeter n.
2
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
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
KEYWORD
nonn
AUTHOR
T. D. Noe, Oct 24 2006
STATUS
approved