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

%I A209916 #17 Sep 08 2022 08:46:01
%S A209916 0,0,2,26,1457,1889567,470184984575,359414999291950792703,
%T A209916 27008149481218253520093899825086463,
%U A209916 12768639440249474099578561928613102801011591209543532543
%N A209916 Kolmogorov's button, 2-color generic convex polygon version.
%C A209916 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.
%C A209916 Kolmogorov's button problem is briefly mentioned in the book by Gessen.
%D A209916 Masha Gessen, Perfect Rigor, A Genius and the Mathematical Breakthrough of the Century, Houghton Mifflin Harcourt, 2009, page 38.
%F A209916 a(n) = A047656(n)*2^A000332(n) - 1.
%e A209916 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.
%t A209916 Table[-1+(3^Binomial[n,2])*(2^Binomial[n,4]),{n,0,9}] (* _Ivan N. Ianakiev_, Dec 29 2015 *)
%o A209916 (Magma) [3^((n^2-n) div 2)*2^Binomial(n,4)-1: n in [0..10]]; // _Vincenzo Librandi_, Dec 29 2015
%Y A209916 Cf. A000332, A047656.
%K A209916 easy,nonn
%O A209916 0,3
%A A209916 _Ivan N. Ianakiev_, Mar 15 2012