A347433 Irregular triangle read by rows: T(n,k) is the difference between the total arch lengths of a semi-meander multiplied by its number of exterior arches and total arch lengths of the semi-meanders with n + 1 top arches generated by the exterior arch splitting algorithm on the given semi-meander.

4, 4, 4, 10, 4, 11, 4, 12, 20, 4, 13, 22, 4, 14, 24, 34, 4, 15, 26, 37, 4, 16, 28, 40, 52, 4, 17, 30, 43, 56, 4, 18, 32, 46, 60, 74, 4, 19, 34, 49, 64, 79, 4, 20, 36, 52, 68, 84, 100, 4, 21, 38, 55, 72, 89, 106, 4, 22, 40, 58, 76, 94, 112, 130, 4, 23, 42, 61, 80, 99

Offset

2,1

Links

Table of n, a(n) for n=2..71.

Formula

For n >= 2 and k = 2..floor((n+2)/2), T(n,k) = 4 + (n+2)*(k-2).

Example

n = number of top arches, k = number of exterior top arches:
n\k  2   3   4   5   6
2:   4
3:   4
4:   4   10
5:   4   11
6:   4   12  20
7:   4   13  22
8:   4   14  24  34
9:   4   15  26  37
10:  4   16  28  40  52
Length of each arch = 1 + number of arches covered:
Top arches of a given semi-meander:       Arch splitting generated
n = 5, k = 2                              semi-meanders (6 top arches):
     1     1    = 2 exterior arches                /\
           /\                                     //\\
     /\   //\\                                   ///\\\
    //\\ ///\\\                           /\ /\ ////\\\\
    21   321    = 9 length of top arches  1  1  4321     = 12 length of top arches
                                            /\
                                           //\\   /\
                                          ///\\\ //\\ /\
                                          321    21   1  = 10 length of top arches
    T(5,2) = 4 + (5+2)(2-2) = 4 --------------------------- 4 = (12+10) - (2 * 9);
Top arches of given semi meander:
n = 5, k = 3                                    /\
    1   1    1   = 3 exterior arches           /  \
        /\   /\                               /    \
    /\ //\\ //\\                             //\  /\\
    1  21   21   = 7 length top arches   /\ ///\\//\\\
                                         1  521  21     = 12 length of top arches
                                                   /\
                                          /\      //\\
                                         //\\ /\ ///\\\
                                         21   1  321    = 10 length of top arches
                                            /\
                                           /  \
                                          /  /\\
                                         //\//\\\ /\ /\
                                         41 21    1  1  = 10 length of top arches
    T(5,3) = 4 + (5+2)(3-2) = 11 --------------------- 11 = (12+10+10) - (3 * 7).

Crossrefs

Cf. A345747.

Sequence in context: A013601 A219802 A219460 * A220010 A190717 A220204

Adjacent sequences: A347430 A347431 A347432 * A347434 A347435 A347436

Keyword

nonn,tabf

Author

Roger Ford, Sep 01 2021

Status

approved