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A366415 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.

%I A366415 #63 Jul 28 2024 09:20:24
%S A366415 10,34,78,222,362,938,1326,3246,4242,10002,12438,28566,34330,77338,
%T A366415 90654,201246,231458,507938,575526,1251366,1400874,3022890,3350574,
%U A366415 7184430,7897138,16842802,18382902,39026742,42336314,89522234,96600126,203554878
%N A366415 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 A366415 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 A366415 Paolo Xausa, <a href="/A366415/b366415.txt">Table of n, a(n) for n = 4..1000</a>
%H A366415 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 A366415 <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 A366415 For n>2:
%F A366415 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 A366415 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 A366415 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 A366415 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 A366415      __                                __
%e A366415     //\\          __        ____      //\\        __          ____
%e A366415    ///\\\    __  //\\      /  /\\    ///\\\      //\\  __    //\  \
%e A366415 /\////\\\\, //\\///\\\, /\//\//\\\, ////\\\\/\, ///\\\//\\, ///\\/\\/\
%e A366415 There are 12 exterior arches for the 6 solutions.
%e A366415 Solutions for generation n+1 using the exterior arch splitting algorithm:
%e A366415       __
%e A366415      //\\                        __                      ____
%e A366415     ///\\\         __           //\\     __             /____\
%e A366415    ////\\\\   __  //\\         ///\\\   //\\  __       //  __\\   __    __
%e A366415 /\/////\\\\\,//\\///\\\/\,/\/\////\\\\,///\\\//\\/\,/\///\//\\\\,//\\/\//\\/\
%e A366415     __
%e A366415    //\\                      __                        ____
%e A366415   ///\\\         __         //\\               __     /____\
%e A366415  ////\\\\       //\\  __   ///\\\         __  //\\   //__  \\       __    __
%e A366415 /////\\\\\/\,/\///\\\//\\,////\\\\/\/\,/\//\\///\\\,////\\/\\\/\,/\//\\/\//\\
%e A366415 These 12 solutions have 34 exterior arches. Therefore a(5) = 34.
%t A366415 LinearRecurrence[{1, 7, -7, -18, 18, 20, -20, -8, 8}, {10, 34, 78, 222, 362, 938, 1326, 3246, 4242}, 50] (* _Paolo Xausa_, May 28 2024 *)
%Y A366415 Cf. A259689, A365679.
%K A366415 nonn,easy
%O A366415 4,1
%A A366415 _Roger Ford_, Oct 10 2023