

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.


10



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
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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 noncongruent 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: kgon partitions, Bull. Austral Math. Soc., 64 (2001), 321329.
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*(1x^11)/((1x)*(1x^2)*(1x^4)*(1x^5)*(1x^6)*(1x^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(1x^11)/((1x)(1x^2)(1x^4)(1x^5)(1x^6) (1x^8)), {x, 0, 60}], x] (* Harvey P. Dale, Dec 16 2011 *)


CROSSREFS

Number of kgons 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



