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A062890 Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.

%I A062890 #50 Jan 02 2021 08:35:35
%S A062890 0,0,0,0,1,1,1,2,3,4,5,7,8,11,12,16,18,23,24,31,33,41,43,53,55,67,69,
%T A062890 83,86,102,104,123,126,147,150,174,177,204,207,237,241,274,277,314,
%U A062890 318,358,362,406,410,458,462,514,519,575,579,640,645,710
%N A062890 Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.
%C A062890 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. - _Frank M Jackson_, Jun 29 2012
%H A062890 Seiichi Manyama, <a href="/A062890/b062890.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A062890 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 A062890 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 A062890 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).
%F A062890 G.f.: x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).
%F A062890 a(2*n+6) = A026810(2*n+6) - A000601(n), a(2*n+7) = A026810(2*n+7) - A000601(n) for n >= 0. - _Seiichi Manyama_, Jun 08 2017
%F A062890 From _Wesley Ivan Hurt_, Jan 01 2021: (Start)
%F A062890 a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-8) + a(n-9) - a(n-10) + a(n-11) + a(n-12) - a(n-13).
%F A062890 a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign(floor((i+j+k)/(n-i-j-k+1))). (End)
%e A062890 a(7) = 2 as the two partitions are (1,2,2,2), (1,1,2,3) and in each sum of any three is more than the fourth.
%t A062890 CoefficientList[Series[x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), {x, 0, 60}], x] (* _Frank M Jackson_, Jun 09 2017 *)
%Y A062890 Number of k-gons that can be formed with perimeter n: A005044 (k=3), this sequence (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).
%K A062890 nonn,easy
%O A062890 0,8
%A A062890 _Amarnath Murthy_, Jun 29 2001
%E A062890 More terms from _Vladeta Jovovic_ and _Dean Hickerson_, Jul 01 2001