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

0, 1, 1, 3, 4, 8, 10, 15, 19, 26, 31, 39, 46, 56, 64, 75, 85, 98, 109, 123, 136, 152, 166, 183, 199, 218

Offset: 3

Author

Kival Ngaokrajang, May 25 2013

Keywords

• nonn
• more

Comments

From the drawings as shown in links, it can be separated the distinct quadrilaterals into 3 types:

Q1: Quadrilaterals which have at least one side equal to n-gon sides length.

Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than n-gon sides length.

Q3: Quadrilaterals which have no parallel sides and all sides are longer than n-gon side length.

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.

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.

Examples

			For a pentagon, there are 5 quadrilaterals which are the same size and shape. Therefore a(5) = 1.
		

Links

• Kival Ngaokrajang, The distinct quadrilaterals for n = 4..9
• Kival Ngaokrajang, The distinct quadrilaterals for n = 10
• Kival Ngaokrajang, Q1 & Q2 for n = 23
• Kival Ngaokrajang, Q3 for n = 23

Crossrefs

Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n-3) is number of distinct triangles in an n-gon.

Formula

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