

A226088


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


2



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
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OFFSET

3,4


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 ngon sides length.
Q2: Quadrilaterals which have at least one pair parallel sides and all sides are longer than ngon sides length.
Q3: Quadrilaterals which have no parallel sides and all sides are longer than ngon side length.
Q1(n) = A004652(n3); Q2(n) = A001917(n6), Q2(3) = 0, Q2(4) = 0; Q3(n) = A005232(n10), 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(n4), for n>=4. Also a(n) = A005232(n4)  A005232(n10), for n>=10.


LINKS

Table of n, a(n) for n=3..28.
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


FORMULA

Empirical g.f.: x^4*(x^2x+1)^2*(x^2+x+1) / ((x1)^3*(x+1)*(x^2+1)).  Colin Barker, Oct 31 2013


EXAMPLE

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


PROG

(Small Basic)
Q2=0
For n = 3 To 50
Q1 = Math.Ceiling((n3)*(n3)/4) 'A004652(n3)
If n > 4 Then
Q2 = Math.Round((n6)*(n6)/8) 'A001917(n6)
EndIf
a = Q1 + Q2
TextWindow.Write(a +", ")
EndFor


CROSSREFS

Cf. A004652, A001917, A005232, A001399: For n >= 3, a(n3) is number of distinct triangles in an ngon.
Sequence in context: A063414 A265611 A310009 * A026494 A043306 A308844
Adjacent sequences: A226085 A226086 A226087 * A226089 A226090 A226091


KEYWORD

nonn,more


AUTHOR

Kival Ngaokrajang, May 25 2013


STATUS

approved



