A124287 Triangle of the number of integer-sided k-gons having perimeter n, for k=3..n.

1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 1, 5, 4, 4, 1, 1, 3, 7, 9, 7, 4, 1, 1, 2, 9, 13, 15, 8, 5, 1, 1, 4, 13, 23, 25, 20, 10, 5, 1, 1, 3, 16, 29, 46, 37, 29, 12, 6, 1, 1, 5, 22, 48, 72, 75, 57, 35, 14, 6, 1, 1, 4, 25, 60, 113, 129, 125, 79, 47, 16, 7, 1, 1, 7, 34, 92, 172, 228, 231, 185

Offset: 3

Author

T. D. Noe, Oct 24 2006

Keywords

• nonn
• tabl

Comments

Rotations and reversals are counted only once. For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. Column k=3 is A005044, column k=4 is A057886, column k=5 is A124285 and column k=6 is A124286. Note that A124278 counts polygons whose sides are nondecreasing.

Examples

			For polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 2 1 1
2 3 3 1 1
1 5 4 4 1 1
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..1277 (terms 3..212 from T. D. Noe)
• James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Crossrefs

Row sums are A293818.

Cf. A293819.

Programs

Mathematica

Needs["DiscreteMath`Combinatorica`"]; Table[p=Partitions[n]; Table[s=Select[p,Length[ # ]==k && #[[1]]Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

PARI

T(n,k)={(sumdiv(gcd(n, k), d, eulerphi(d)*binomial(n/d, k/d))/n + binomial(k\2 + (n-k)\2, k\2) - binomial(n\2, k-1) - binomial(n\4, k\2) - if(k%2, 0, binomial((n+2)\4, k\2)))/2;}
for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 21 2017

Formula

A formula is given in Theorem 1.5 of the East and Niles article.