OFFSET
7,1
COMMENTS
The densest possible star-shaped regular n-gon is formed by connecting with straight lines every a(n)-th point out of n regularly spaced points lying on a circumference.
For a given n there are A055684(n) different star-shaped regular polygons. The minimum skip increment for connecting points on the circumference is given by A053669(n), the maximum skip increment is given by a(n). There are no star-shaped polygons for n=3,4,6 and unique star-shaped polygons for n=5,8,10 and 12, for which a(n) = A053669(n).
LINKS
Colin Barker, Table of n, a(n) for n = 7..1000
Jay Kappraff, Gary W. Adamson, Polygons and Chaos, BRIDGES Mathematical Connections in Art, Music, and Science, 2001.
Hugo Pfoertner, Star-shaped regular polygons.
Hugo Pfoertner, (Star-Shaped-) Polygons with Maximal Density.
Eric Weisstein's World of Mathematics, Star Polygon.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
a(4*k-1) = a(4*k) = a(4*k+2) = 2*k-1; a(4*k+1) = 2*k.
a(n) = (1/2) (n - (I^n + (-I)^n)/2 - (-1)^n + 4). - Ralf Stephan, May 17 2007
a(n) = a(n-1)+a(n-4)-a(n-5) for n>11. - Colin Barker, Feb 21 2015
G.f.: -x^7*(x^4+x^3-x^2-3) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 21 2015
MATHEMATICA
lnc[n_]:=Module[{k=Floor[n/2]}, While[!CoprimeQ[n, k], k--]; k]; Array[ lnc, 90, 7] (* Harvey P. Dale, May 15 2021 *)
PROG
(PARI) Vec(-x^7*(x^4+x^3-x^2-3)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 21 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Hugo Pfoertner, Jan 23 2005
STATUS
approved