A339179 -id:A339179 - cp's OEIS frontend

A354387 a(n) is the number of arch configuration solutions with n arches derived from 2 concentric arches using the exterior arch splitting algorithm.

1, 1, 3, 6, 18, 42, 130, 332, 1048, 2836, 9078, 25578, 82730, 240124, 782956, 2324800

Offset: 2

Roger Ford, May 24 2022

Every a(n) arch configuration solution when put on top of n concentric arches has exactly 2 components (2 distinct loops).
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.
For a(n) with n odd, n > 2 and a center arch of /\, a(n) = A000682(n-1).
There is an infinite number of the starting arch configurations with one exterior arch. They generate an infinite number of unique sequences.
Conjecture from Roger Ford, Aug 26 2022: (Start)
a(n) = a subset of semi-meanders A000682(n+1) with an arch of length 1 starting in the second top arch position.
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.
Solutions:     /\                                 /\
              /  \           /\   /\        /\   //\\
             //\/\\ /\/\,   //\\ //\\ /\,  //\\ ///\\\
Nonsolutions:               /\          /\
         /\                //\\        //\\
        //\\   /\         ///\\\      ///\\\            /\   /\
       ///\\\ //\\,   /\ ////\\\\,   ////\\\\ /\,   /\ //\\ //\\
         /\               /\
        /  \             /  \             /\
       //\  \           /  /\\           /  \
      ///\\/\\ /\,  /\ //\//\\\,  /\ /\ //\/\\  (End)

			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.
                                              split
      split                        split       /\                       /\
       /\         split        /\   /\        //\\                /\   /  \
      //\\  =>  /\ /\ /\  =>  //\\ //\\  =>  ///\\\ /\ /\  => /\ //\\ //\/\\
arches 2           3             4                5                 6

Cf. A000682, A331499, A339179, A287548.

a(n) = A331499(n, 2).
Conjecture: For n >= 2, a(n) = Sum_{k = 2..floor((n+2)/2)} A339179 T(n,k)*(k-1).
a(n) = A287548(n, n-1) - A287548(n, n).

cp's OEIS Frontend

A354387 a(n) is the number of arch configuration solutions with n arches derived from 2 concentric arches using the exterior arch splitting algorithm.

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