A105638 Maximum number of intersections in self-intersecting n-gon.

0, 1, 5, 7, 14, 17, 27, 31, 44, 49, 65, 71, 90, 97, 119, 127, 152, 161, 189, 199, 230, 241, 275, 287, 324, 337, 377, 391, 434, 449, 495, 511, 560, 577, 629, 647, 702, 721, 779, 799, 860, 881, 945, 967, 1034, 1057, 1127, 1151, 1224, 1249, 1325, 1351, 1430, 1457

Offset: 3

Author

David W. Wilson, Apr 16 2005

Keywords

• nonn
• easy

Comments

Quasipolynomial of order 2. - Charles R Greathouse IV, Mar 29 2012

Examples

			The self-intersecting pentagon with the largest number of intersections is the star polygon {5/2} (pentagram), with 5 intersections, hence a(5) = 5.
		

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).

Programs

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 *)

PARI

a(n)=if(n%2,n*(n-3)/2,n*(n-4)/2+1) \\ Charles R Greathouse IV, Mar 29 2012

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