

A209916


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


0




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 ngon, i.e., a convex ngon 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 ngon, equaling 2^A000332(n), is taken into account (as reddiagonalovergreendiagonal, for instance, is a different pattern from greendiagonaloverreddiagonal). In general, if the number of colors is c, then a(n) = ((c+1)^(n1)*n/2)*((c1)*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.


LINKS

Table of n, a(n) for n=0..9.


FORMULA

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


EXAMPLE

For the classic 4hole 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.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A000332, A047656.
Sequence in context: A002704 A015215 A158120 * A337578 A156213 A159318
Adjacent sequences: A209913 A209914 A209915 * A209917 A209918 A209919


KEYWORD

easy,nonn


AUTHOR

Ivan N. Ianakiev, Mar 15 2012


STATUS

approved



