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A124285
Number of integer-sided pentagons having perimeter n.
2
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
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: A377040 A167930 A326981 * A131326 A089300 A250111
KEYWORD
nonn
AUTHOR
T. D. Noe, Oct 24 2006
STATUS
approved