A365679 - OEIS

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

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OFFSET

2,1

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.

LINKS

Table of n, a(n) for n=2..37.

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

Cf. A259689.

Sequence in context: A022382 A162521 A125302 * A175588 A239788 A136862

Adjacent sequences: A365676 A365677 A365678 * A365680 A365681 A365682

KEYWORD

nonn

AUTHOR

Roger Ford, Sep 15 2023

EXTENSIONS

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

STATUS

approved