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A226088 a(n) is the number of the distinct quadrilaterals in a regular n-gon, 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 (list; graph; refs; listen; history; text; internal format)
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 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.

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^2-x+1)^2*(x^2+x+1) / ((x-1)^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((n-3)*(n-3)/4) 'A004652(n-3)

  If n > 4 Then

    Q2 = Math.Round((n-6)*(n-6)/8) 'A001917(n-6)

  EndIf

  a = Q1 + Q2

  TextWindow.Write(a +", ")

EndFor

CROSSREFS

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

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

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)