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A366415 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) exterior top arches using the exterior arch splitting algorithm. 0
10, 34, 78, 222, 362, 938, 1326, 3246, 4242, 10002, 12438, 28566, 34330, 77338, 90654, 201246, 231458, 507938, 575526, 1251366, 1400874, 3022890, 3350574, 7184430, 7897138, 16842802, 18382902, 39026742, 42336314, 89522234, 96600126, 203554878 (list; graph; refs; listen; history; text; internal format)
OFFSET
4,1
COMMENTS
b(n) = ((n-4)*2^floor((n-1)/2)+2)*floor(n/2) is the number of exterior top arches for all semi-meander solutions with n top arches and floor(n/2) exterior top arches. Conjecture: for n>=5, lim_{n->oo} a(n)/b(n) = 3.
LINKS
Michael LaCroix, Approaches to the Enumerative Theory of Meanders, 2003, pg. 31-31, Demonstrates arch splitting with semi-meander models.
FORMULA
For n>2:
a(2*n) = (3*n-1)*((2*n-4)*2^(n-1) + 2) - (3*n-3)*((2*n-5)*2^(n-1) + 2) + a(2*n-1);
a(2*n+1) = 3*n*((2*n-3)*2^n + 2) - 3*n*((2*n-4)*2^(n-1) + 2) + a(2*n).
G.f.: 2*x^4*(5 + 12*x - 13*x^2 - 12*x^3 + 6*x^4)/((1 - x)^2*(1 + x)*(1 - 2*x^2)^3). - Stefano Spezia, Nov 07 2023
EXAMPLE
For n=5, the number of semi-meanders with 5 top arches and 2 exterior top arches is equal to A259689(5,2) = 6:
__ __
//\\ __ ____ //\\ __ ____
///\\\ __ //\\ / /\\ ///\\\ //\\ __ //\ \
/\////\\\\, //\\///\\\, /\//\//\\\, ////\\\\/\, ///\\\//\\, ///\\/\\/\
There are 12 exterior arches for the 6 solutions.
Solutions for generation n+1 using the exterior arch splitting algorithm:
__
//\\ __ ____
///\\\ __ //\\ __ /____\
////\\\\ __ //\\ ///\\\ //\\ __ // __\\ __ __
/\/////\\\\\,//\\///\\\/\,/\/\////\\\\,///\\\//\\/\,/\///\//\\\\,//\\/\//\\/\
__
//\\ __ ____
///\\\ __ //\\ __ /____\
////\\\\ //\\ __ ///\\\ __ //\\ //__ \\ __ __
/////\\\\\/\,/\///\\\//\\,////\\\\/\/\,/\//\\///\\\,////\\/\\\/\,/\//\\/\//\\
These 12 solutions have 34 exterior arches. Therefore a(5) = 34.
CROSSREFS
Sequence in context: A155486 A225276 A008527 * A007584 A218329 A009924
KEYWORD
nonn,easy
AUTHOR
Roger Ford, Oct 10 2023
STATUS
approved

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Last modified April 22 02:54 EDT 2024. Contains 371887 sequences. (Running on oeis4.)