A069906 - OEIS

%I #41 Jun 08 2017 08:23:47

%S 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,

%T 150,184,197,239,256,306,325,385,409,480,507,590,623,719,756,867,911,

%U 1038,1087,1232,1289,1453,1516,1701,1774,1981,2061,2293

%N 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.

%C From _Frank M Jackson_, Jul 10 2012: (Start)

%C I recently commented on A062890 that:

%C "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."

%C 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:

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

%H Seiichi Manyama, <a href="/A069906/b069906.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.uni-linz.ac.at/research/combinat/risc/publications/#ppaule">MacMahon's partition analysis III. The Omega package</a>, p. 19.

%H G. E. Andrews, P. Paule and A. Riese, <a href="http://www.risc.jku.at/publications/download/risc_163/PAIX.pdf">MacMahon's Partition Analysis IX: k-gon partitions</a>, Bull. Austral Math. Soc., 64 (2001), 321-329.

%H <a href="/index/Rec#order_25">Index entries for linear recurrences with constant coefficients</a>, 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).

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

%F 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

%t 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 *)

%Y 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).

%K nonn,easy

%O 0,8

%A _N. J. A. Sloane_, May 05, 2002