OFFSET
3,3
COMMENTS
Quasipolynomial of order 2. [Charles R Greathouse IV, Mar 29 2012]
REFERENCES
B. Grünbaum, Selfintersections of Polygons, Geombinatorics, Volume VIII 4 (1998), pp. 37-45.
LINKS
David W. Wilson, Table of n, a(n) for n = 3..10000
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).
FORMULA
a(n) = n(n-3)/2 if n odd, n(n-4)/2+1 if n even.
a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5).
G.f.: x^4*(1+4*x-x^3)/((1+x)^2*(1-x)^3). [Colin Barker, Jan 31 2012]
EXAMPLE
The self-intersecting pentagon with the largest number of intersections is the star polygon {5/2} (pentacle}, with 5 intersections, hence a(5) = 5.
MATHEMATICA
LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 5, 7, 14}, 54] (* or *)
DeleteCases[CoefficientList[Series[x^4*(1 + 4 x - x^3)/((1 + x)^2*(1 - x)^3), {x, 0, 56}], x], 0] (* Michael De Vlieger, Jul 10 2020 *)
PROG
(PARI) a(n)=if(n%2, n*(n-3)/2, n*(n-4)/2+1) \\ Charles R Greathouse IV, Mar 29 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David W. Wilson, Apr 16 2005
STATUS
approved