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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
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.
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
KEYWORD
nonn,more
AUTHOR
Kival Ngaokrajang, May 25 2013
STATUS
approved