A346757 a(n) is the number of arches with a length of one and exactly one covering arch for semi-meanders with n top arches.

0, 2, 2, 12, 20, 88, 180, 716, 1648, 6244, 15512, 57188, 149892, 543324, 1481636, 5310764, 4930380, 53102508, 152935148, 540918660, 1588618212, 5595773136, 16701973552

Offset

2,2

Comments

The ratio of the number of semi-meanders in generation n+1 to the number of semi-meanders in generation n is equal to the ratio of the differences of these semi-meander subset arch sequences in successive generations.

A000682(n+1)/A000682(n) = (A260785(n+3) - a(n+1))/(A260785(n+2) - a(n)).

References

See A000682.

Links

Table of n, a(n) for n=2..24.

P. Di Francesco, O. Golinelli and E. Guitter, Meanders: exact asymptotics, arXiv:cond-mat/9910453 [cond-mat.stat-mech], 1999-2000; Nuclear Physics B, volume 570, issue 3, 27 March 2000, 699-712.

Formula

For n >= 2, a(n) = A260785(n+2) - 2*A000682(n).

Example

n = 5, || indicates an arch of length one with exactly one covering arch.
           /\       /\                                                /\
     /\   //\\     //\\   /\        /\   /\     /\   /\              /  \
    //\\ ///\\\   ///\\\ //\\   /\ //\\ //\\   //\\ //\\ /\   /\ /\ //\/\\
     ||                   ||        ||   ||     ||   ||              ||||
       /\               /\
      /  \             /  \       /\
     //\  \           /  /\\     /  \
    ///\\/\\ /\   /\ //\//\\\   //\/\\ /\ /\
         ||           ||         ||||            a(5) = 12.

Crossrefs

Cf. A000682, A260785.

Sequence in context: A054916 A194767 A302368 * A202669 A178845 A140431

Adjacent sequences: A346754 A346755 A346756 * A346758 A346759 A346760

Keyword

nonn,more

Author

Roger Ford, Aug 01 2021

Status

approved