A057886 -id:A057886 - cp's OEIS frontend

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

T. D. Noe, Oct 24 2006

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.

			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

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.

Row sums are A293818.

Cf. A293819.

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

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

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

A057887 Number of 4-tuples of integers with GCD=1 and giving the lengths of sides of a nondegenerate quadrilateral with perimeter n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 4, 7, 8, 13, 13, 22, 22, 33, 33, 50, 45, 70, 65, 92, 87, 125, 111, 160, 145, 196, 184, 252, 215, 308, 278, 359, 330, 440, 385, 525, 464, 593, 546, 715, 606, 825, 735, 905, 832, 1078, 926, 1219, 1065, 1328, 1223, 1547, 1310, 1715, 1529, 1855

Offset: 1

John W. Layman, Sep 19 2000

nonn

			There are five quadrilaterals with perimeter 8, with sides (1,1,3,3), (1,2,2,3), (1,2,3,2), (1,3,1,3) and (2,2,2,2). (2,2,2,2) is omitted since it has GCD=2, so a(8)=4.

Cf. A051493.

Moebius transform of A057886.

A124285 Number of integer-sided pentagons having perimeter n.

Original entry on oeis.org

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

T. D. Noe, Oct 24 2006

nonn

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

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

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

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

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==5 && #[[1]]

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

A124286 Number of integer-sided hexagons having perimeter n.

Original entry on oeis.org

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

T. D. Noe, Oct 24 2006

nonn

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

			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).

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

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

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==6 && #[[1]]

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A057887 Number of 4-tuples of integers with GCD=1 and giving the lengths of sides of a nondegenerate quadrilateral with perimeter n.

Views

Author

Keywords

Examples

Crossrefs

Formula

A124285 Number of integer-sided pentagons having perimeter n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A124286 Number of integer-sided hexagons having perimeter n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula