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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.
1
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
2,1
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.
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
KEYWORD
nonn
AUTHOR
Roger Ford, Sep 15 2023
EXTENSIONS
a(30) corrected by Georg Fischer, Jun 03 2024
STATUS
approved