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A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.

%I A226088 #26 Feb 11 2018 03:07:26
%S A226088 0,1,1,3,4,8,10,15,19,26,31,39,46,56,64,75,85,98,109,123,136,152,166,
%T A226088 183,199,218
%N A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, which Q3 type are excluded.
%C A226088 From the drawings as shown in links, it can be separated the distinct quadrilaterals into 3 types:
%C A226088 Q1: Quadrilaterals which have at least one side equal to n-gon sides length.
%C A226088 Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than n-gon sides length.
%C A226088 Q3: Quadrilaterals which have no parallel sides and all sides are longer than n-gon side length.
%C A226088 Q1(n) = A004652(n-3); Q2(n) = A001917(n-6), Q2(3) = 0, Q2(4) = 0; Q3(n) = A005232(n-10), Q3(3) = 0, Q3(4) = 0, Q3(5) = 0, Q3(6) = 0, Q3(7) = 0, Q3(8) = 0, Q3(9) = 0.
%C A226088 a(n) = Q1(n) + Q2(n). The total distinct quadrilaterals is Q1 + Q2 + Q3. Also the total distinct quadrilaterals = A005232(n-4), for n>=4. Also a(n) = A005232(n-4) - A005232(n-10), for n>=10.
%H A226088 Kival Ngaokrajang, <a href="/A226088/a226088.jpg">The distinct quadrilaterals for n = 4..9</a>
%H A226088 Kival Ngaokrajang, <a href="/A226088/a226088_2.jpg">The distinct quadrilaterals for n = 10</a>
%H A226088 Kival Ngaokrajang, <a href="/A226088/a226088_3.jpg">Q1 & Q2 for n = 23</a>
%H A226088 Kival Ngaokrajang, <a href="/A226088/a226088_4.jpg">Q3 for n = 23</a>
%F A226088 Empirical g.f.: -x^4*(x^2-x+1)^2*(x^2+x+1) / ((x-1)^3*(x+1)*(x^2+1)). - _Colin Barker_, Oct 31 2013
%e A226088 For a pentagon, there are 5 quadrilaterals which are the same size and shape. Therefore a(5) = 1.
%o A226088 (Small Basic)
%o A226088 Q2=0
%o A226088 For n = 3 To 50
%o A226088   Q1 = Math.Ceiling((n-3)*(n-3)/4) 'A004652(n-3)
%o A226088   If n > 4 Then
%o A226088     Q2 = Math.Round((n-6)*(n-6)/8) 'A001917(n-6)
%o A226088   EndIf
%o A226088   a = Q1 + Q2
%o A226088   TextWindow.Write(a +", ")
%o A226088 EndFor
%Y A226088 Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n-3) is number of distinct triangles in an n-gon.
%K A226088 nonn,more
%O A226088 3,4
%A A226088 _Kival Ngaokrajang_, May 25 2013