A365679 a(n) is the number of exterior top arches (no covering arch) for semi-meanders in generation n+1 that are generated by semi-meanders with n top arches and floor((n+2)/2) exterior top arches using the exterior arch splitting algorithm.

4, 10, 14, 32, 40, 88, 104, 224, 256, 544, 608, 1280, 1408, 2944, 3200, 6656, 7168, 14848, 15872, 32768, 34816, 71680, 75776, 155648, 163840, 335872, 352256, 720896, 753664, 1540096, 1605632, 3276800, 3407872, 6946816, 7208960, 14680064

Offset: 2

Author

Roger Ford, Sep 15 2023

Keywords

• nonn

Comments

b(n) = the number of exterior top arches for all semi-meanders with n top arches and floor((n+2)/2) exterior top arches = (floor(n/2)+1) * 2^(floor((n-1)/2)). For n>=2, lim_{n->oo} a(n)/b(n) = 3.

Examples

			For n=4, the number of semi-meanders with 4 top arches and 3 exterior top arches is equal to A259689(4,3) = 2:
       /\    /\
/\ /\ //\\, //\\ /\ /\ = 6 exterior arches. These 6 arches will generate 6 solutions in the n+1 generation using the exterior arch splitting algorithm.
     _____           __                                    __           _____
    /   /\\    /\   //\\    _____                _____    //\\   /\    //\   \
/\ //\ //\\\, //\\ ///\\\, //\ /\\ /\ /\, /\ /\ //\ /\\, ///\\\ //\\, ///\\ /\\ /\
These 6 solutions have 14 exterior arches. Therefore a(4) = 14.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -4).

Crossrefs

Cf. A259689.

Programs

Mathematica

a[n_]:=(2*n-Floor[(n-1)/2]) * 2^Floor[(n-1)/2]; Array[a,36,2] (* Stefano Spezia, Sep 16 2023 *)

Formula

a(n) = (2*n-floor((n-1)/2)) * 2^floor((n-1)/2).

Extensions

a(30) corrected by Georg Fischer, Jun 03 2024