|
%I #54 Dec 17 2023 11:21:22
%S 10,34,78,222,362,938,1326,3246,4242,10002,12438,28566,34330,77338,
%T 90654,201246,231458,507938,575526,1251366,1400874,3022890,3350574,
%U 7184430,7897138,16842802,18382902,39026742,42336314,89522234,96600126,203554878
%N a(n) is the number of exterior top arches (no covering arch) for semi-meanders in generation n+1 that are generated by semi-meanders with n top arches and floor(n/2) exterior top arches using the exterior arch splitting algorithm.
%C b(n) = ((n-4)*2^floor((n-1)/2)+2)*floor(n/2) is the number of exterior top arches for all semi-meander solutions with n top arches and floor(n/2) exterior top arches. Conjecture: for n>=5, lim_{n->oo} a(n)/b(n) = 3.
%H Michael LaCroix, <a href="https://www.math.uwaterloo.ca/~malacroi/Latex/Meanders.pdf">Approaches to the Enumerative Theory of Meanders</a>, 2003, pg. 31-31, Demonstrates arch splitting with semi-meander models.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,7,-7,-18,18,20,-20,-8,8).
%F For n>2:
%F a(2*n) = (3*n-1)*((2*n-4)*2^(n-1) + 2) - (3*n-3)*((2*n-5)*2^(n-1) + 2) + a(2*n-1);
%F a(2*n+1) = 3*n*((2*n-3)*2^n + 2) - 3*n*((2*n-4)*2^(n-1) + 2) + a(2*n).
%F G.f.: 2*x^4*(5 + 12*x - 13*x^2 - 12*x^3 + 6*x^4)/((1 - x)^2*(1 + x)*(1 - 2*x^2)^3). - _Stefano Spezia_, Nov 07 2023
%e For n=5, the number of semi-meanders with 5 top arches and 2 exterior top arches is equal to A259689(5,2) = 6:
%e __ __
%e //\\ __ ____ //\\ __ ____
%e ///\\\ __ //\\ / /\\ ///\\\ //\\ __ //\ \
%e /\////\\\\, //\\///\\\, /\//\//\\\, ////\\\\/\, ///\\\//\\, ///\\/\\/\
%e There are 12 exterior arches for the 6 solutions.
%e Solutions for generation n+1 using the exterior arch splitting algorithm:
%e __
%e //\\ __ ____
%e ///\\\ __ //\\ __ /____\
%e ////\\\\ __ //\\ ///\\\ //\\ __ // __\\ __ __
%e /\/////\\\\\,//\\///\\\/\,/\/\////\\\\,///\\\//\\/\,/\///\//\\\\,//\\/\//\\/\
%e __
%e //\\ __ ____
%e ///\\\ __ //\\ __ /____\
%e ////\\\\ //\\ __ ///\\\ __ //\\ //__ \\ __ __
%e /////\\\\\/\,/\///\\\//\\,////\\\\/\/\,/\//\\///\\\,////\\/\\\/\,/\//\\/\//\\
%e These 12 solutions have 34 exterior arches. Therefore a(5) = 34.
%Y Cf. A259869, A365679.
%K nonn,easy
%O 4,1
%A _Roger Ford_, Oct 10 2023
|
