OFFSET
0,3
COMMENTS
This sequence shows the number of distinct patterns that can be created with threads of 2 colors while sewing on a button with n buttonholes located on the vertices of a generic convex n-gon, i.e., a convex n-gon with no more than two diagonals intersecting at any point in its interior. The number of all distinct patterns due to intersections made by differently colored diagonals of the n-gon, equaling 2^A000332(n), is taken into account (as red-diagonal-over-green-diagonal, for instance, is a different pattern from green-diagonal-over-red-diagonal). In general, if the number of colors is c, then a(n) = ((c+1)^(n-1)*n/2)*((c-1)*c)^A000332(n)-1.
Kolmogorov's button problem is briefly mentioned in the book by Gessen.
REFERENCES
Masha Gessen, Perfect Rigor, A Genius and the Mathematical Breakthrough of the Century, Houghton Mifflin Harcourt, 2009, page 38.
EXAMPLE
MATHEMATICA
Table[-1+(3^Binomial[n, 2])*(2^Binomial[n, 4]), {n, 0, 9}] (* Ivan N. Ianakiev, Dec 29 2015 *)
PROG
(Magma) [3^((n^2-n) div 2)*2^Binomial(n, 4)-1: n in [0..10]]; // Vincenzo Librandi, Dec 29 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ivan N. Ianakiev, Mar 15 2012
STATUS
approved