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

2,3

Comments

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)

Links

Table of n, a(n) for n=2..17.

Formula

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).

Example

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

Crossrefs

Cf. A000682, A331499, A339179, A287548.

Sequence in context: A342583 A007990 A197050 * A121188 A331678 A120718

Adjacent sequences: A354384 A354385 A354386 * A354388 A354389 A354390

Keyword

nonn,more

Author

Roger Ford, May 24 2022

Status

approved