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

Author

Roger Ford, Aug 01 2021

Keywords

• nonn
• more

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)).

Examples

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

References

• See A000682.

Links

• 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.

Crossrefs

Cf. A000682, A260785.

Formula

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