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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. 6
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 (list; graph; refs; listen; history; text; internal format)
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

T. D. Noe, Table of n, a(n) for n = 0..1000

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.

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)).

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

Cf. A005044, A062890, A069907, A062890.

Sequence in context: A238494 A267046 A166515 * A183564 A222707 A053097

Adjacent sequences:  A069903 A069904 A069905 * A069907 A069908 A069909

KEYWORD

nonn,easy

AUTHOR

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

STATUS

approved

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Last modified April 28 23:30 EDT 2017. Contains 285604 sequences.