A209916 Kolmogorov's button, 2-color generic convex polygon version.

0, 0, 2, 26, 1457, 1889567, 470184984575, 359414999291950792703, 27008149481218253520093899825086463, 12768639440249474099578561928613102801011591209543532543

Offset: 0

Author

Ivan N. Ianakiev, Mar 15 2012

Keywords

• easy
• nonn

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.

Examples

			For the classic 4-hole button (where n=4 and c=2) the number of distinct patterns is a(n) = A047656(4)*2^A000332(4) - 1 = 729*2 - 1 = 1457. The "-1" stands for the case where the threads are missing, i.e., where the button is unattached to the cloth.
		

References

• Masha Gessen, Perfect Rigor, A Genius and the Mathematical Breakthrough of the Century, Houghton Mifflin Harcourt, 2009, page 38.

Crossrefs

Cf. A000332, A047656.

Programs

Magma

[3^((n^2-n) div 2)*2^Binomial(n,4)-1: n in [0..10]]; // Vincenzo Librandi, Dec 29 2015

Mathematica

Table[-1+(3^Binomial[n,2])*(2^Binomial[n,4]),{n,0,9}] (* Ivan N. Ianakiev, Dec 29 2015 *)

Formula

a(n) = A047656(n)*2^A000332(n) - 1.