This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354387 #41 Aug 31 2022 02:42:49
%S A354387 1,1,3,6,18,42,130,332,1048,2836,9078,25578,82730,240124,782956,
%T A354387 2324800
%N A354387 a(n) is the number of arch configuration solutions with n arches derived from 2 concentric arches using the exterior arch splitting algorithm.
%C A354387 Every a(n) arch configuration solution when put on top of n concentric arches has exactly 2 components (2 distinct loops).
%C A354387 If the starting arch configuration is /\, the exterior splitting algorithm will generate all the top arch configurations for semi-meanders. A000682(n) is the number of semi-meanders with n top arches.
%C A354387 For a(n) with n odd, n > 2 and a center arch of /\, a(n) = A000682(n-1).
%C A354387 There is an infinite number of the starting arch configurations with one exterior arch. They generate an infinite number of unique sequences.
%C A354387 Conjecture from _Roger Ford_, Aug 26 2022: (Start)
%C A354387 a(n) = a subset of semi-meanders A000682(n+1) with an arch of length 1 starting in the second top arch position.
%C A354387 Example: a(4) = 3, There are 10 semi-meanders with 5 top arches. 3 of those semi-meanders have an arch of length 1 starting in the second position.
%C A354387 Solutions: /\ /\
%C A354387 / \ /\ /\ /\ //\\
%C A354387 //\/\\ /\/\, //\\ //\\ /\, //\\ ///\\\
%C A354387 Nonsolutions: /\ /\
%C A354387 /\ //\\ //\\
%C A354387 //\\ /\ ///\\\ ///\\\ /\ /\
%C A354387 ///\\\ //\\, /\ ////\\\\, ////\\\\ /\, /\ //\\ //\\
%C A354387 /\ /\
%C A354387 / \ / \ /\
%C A354387 //\ \ / /\\ / \
%C A354387 ///\\/\\ /\, /\ //\//\\\, /\ /\ //\/\\ (End)
%F A354387 a(n) = A331499(n, 2).
%F A354387 Conjecture: For n >= 2, a(n) = Sum_{k = 2..floor((n+2)/2)} A339179 T(n,k)*(k-1).
%F A354387 a(n) = A287548(n, n-1) - A287548(n, n).
%e A354387 The splitting exterior arch algorithm involves splitting an exterior arch and moving the split ends to the first and last position of the arch configuration on the x axis. Moving the ends of the split arch will cause one arch to disappear and two new arches to appear. The example below shows one exterior arch being split in a generation.
%e A354387 split
%e A354387 split split /\ /\
%e A354387 /\ split /\ /\ //\\ /\ / \
%e A354387 //\\ => /\ /\ /\ => //\\ //\\ => ///\\\ /\ /\ => /\ //\\ //\/\\
%e A354387 arches 2 3 4 5 6
%Y A354387 Cf. A000682, A331499, A339179, A287548.
%K A354387 nonn,more
%O A354387 2,3
%A A354387 _Roger Ford_, May 24 2022