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.

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

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

Table of n, a(n) for n=4..35.

Michael LaCroix, Approaches to the Enumerative Theory of Meanders, 2003, pg. 31-31, Demonstrates arch splitting with semi-meander models.

Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-18,18,20,-20,-8,8).

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

Cf. A259869, A365679.

Sequence in context: A155486 A225276 A008527 * A007584 A218329 A009924

Adjacent sequences: A366412 A366413 A366414 * A366416 A366417 A366418

Keyword

nonn,easy

Author

Roger Ford, Oct 10 2023

Status

approved