A069906 Number of pentagons that can be formed with perimeter n. In other words, number of partitions of n into five parts such that the sum of any four is more than the fifth.

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 23, 25, 35, 39, 52, 57, 74, 81, 103, 111, 139, 150, 184, 197, 239, 256, 306, 325, 385, 409, 480, 507, 590, 623, 719, 756, 867, 911, 1038, 1087, 1232, 1289, 1453, 1516, 1701, 1774, 1981, 2061, 2293

Offset

0,8

Comments

From Frank M Jackson, Jul 10 2012: (Start)

I recently commented on A062890 that:

"Partition sets of n into four parts (sides) such that the sum of any three is more than the fourth do not uniquely define a quadrilateral, even if it is further constrained to be cyclic. This is because the order of adjacent sides is important. E.g. the partition set [1,1,2,2] for a perimeter n=6 can be reordered to generate two non-congruent cyclic quadrilaterals, [1,2,1,2] and [1,1,2,2], where the first is a rectangle and the second a kite."

This comment applies to all integer polygons (other than triangles) that are generated from a perimeter of length n. Not sure how best to correct for the above observation but my suggestion would be to change the definition of the present sequence to read:

"The number of cyclic integer pentagons differing only in circumradius that can be generated from an integer perimeter n." (End)

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.

G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.

Index entries for linear recurrences with constant coefficients, signature (0, 1, 0, 1, 1, 0, -1, 0, -1, -2, 0, 0, 0, 0, 2, 1, 0, 1, 0, -1, -1, 0, -1, 0, 1).

Formula

G.f.: x^5*(1-x^11)/((1-x)*(1-x^2)*(1-x^4)*(1-x^5)*(1-x^6)*(1-x^8)).

a(2*n+8) = A026811(2*n+8) - A002621(n), a(2*n+9) = A026811(2*n+9) - A002621(n) for n >= 0. - Seiichi Manyama, Jun 08 2017

Mathematica

CoefficientList[Series[x^5(1-x^11)/((1-x)(1-x^2)(1-x^4)(1-x^5)(1-x^6) (1-x^8)), {x, 0, 60}], x] (* Harvey P. Dale, Dec 16 2011 *)

Crossrefs

Number of k-gons that can be formed with perimeter n: A005044 (k=3), A062890 (k=4), this sequence (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

Sequence in context: A326524 A326629 A326445 * A304332 A183564 A222707

Adjacent sequences: A069903 A069904 A069905 * A069907 A069908 A069909

Keyword

nonn,easy

Author

N. J. A. Sloane, May 05, 2002

Status

approved