OFFSET
0,1
COMMENTS
We consider n points in the plane, {A1, A2, ..., An}, n > = 4, lying on a line; a(n) is the number of points of intersections of the circles with diameters AiAj (i<>j).
We consider n aligned points of the plane {A1, A2, ..., An}, n > = 4. The sequence a(n) is the number of points of intersections of the circles with diameters AiAj (i<>j). - Michel Lagneau, May 04 2010
REFERENCES
J. M. Monier. Algèbre & Géometrie, Dunod 1996. Exercise p. 62.
LINKS
Michel Lagneau, Illustration for n=4
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = (n^4 + 10n^3 + 35n^2 + 62n + 12)/12. - Gary Detlefs, Jun 06 2010
G.f.: -(3*x^4-13*x^3+21*x^2-15*x+6) / (x-1)^5. - Colin Barker, Dec 20 2012
EXAMPLE
For n = 5, we obtain 2*5 + 5 = 15 intersections.
From Michel Lagneau, May 04 2010: (Start)
For n = 4, we obtain 2*1 + 4 = 6 intersections (including tangential circles);
For n = 5, we obtain 2*5 + 5 = 15 intersections (including tangential circles). (End)
MAPLE
with(numtheory): n0:=75: T:=array(1..n0-4):for n from 5 to n0 do: T[n-4]:= 2*binomial(n, 4)+n:od:print(T):
with(numtheory): n0:=75: T:=array(1..n0-3):for n from 4 to n0 do: T[n-3]:= 2*binomial(n, 4)+n:od:print(T): # Michel Lagneau, May 04 2010
seq(2*binomial(n+4, 4)+n+4, n=0..39); # Gary Detlefs, Jun 06 2010
MATHEMATICA
2Binomial[#+4, 4]+#+4&/@Range[0, 40] (* Harvey P. Dale, Feb 08 2011 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michel Lagneau, May 04 2010
EXTENSIONS
Definition corrected by Gary Detlefs, Jun 06 2010
STATUS
approved