A363910 Triangular array read by rows: T(n,k) = the number of closed meanders with n top arches and k closed meanders in the reduction of the closed meander by the reverse of the exterior arch splitting algorithm.

1, 0, 2, 0, 2, 6, 0, 6, 14, 22, 0, 28, 56, 86, 92, 0, 162, 298, 428, 518, 422, 0, 1076, 1868, 2562, 3096, 3144, 2074, 0, 7852, 13076, 17292, 20624, 21990, 19366, 10754

Offset: 1

Author

Roger Ford, Jun 27 2023

Keywords

• nonn
• tabl
• more

Comments

The terms of this sequence can also be derived from sequences of consecutively numbered stamps folded with stamp 1 on top.

Examples

			n\k  1     2      3     4     5     6     7     8
1:   1
2:   0     2
3:   0     2      6
4:   0     6     14    22
5:   0    28     56    86    92
6:   0   162    298   428   518   422
7:   0  1076   1868  2562  3096  3144  2074
8:   0  7852  13076 17292 20624 21990 19366 10754
Closed meander:         Closed meander split with bottom rotated right
4 top arches            to form top of semi-meander with 8 arches
    ______                   ______
   / ____ \                 / ____ \
  / / __ \ \               / / __ \ \              __
 / / /  \ \ \             / / /  \ \ \            /  \
/ / / /\ \ \ \           / / / /\ \ \ \  /\  /\  / /\ \
\ \/ /  \/  \/           binary representation of semi-meander
 \__/                    1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0
                    Semi-meander top arches with no covering center arch  =  cm
                              START:          center |
Reduction of semi-meander:    1  1  1  1  0  0  0  0  1  0  1  0  1  1  0  0 cm(1)
Combine end of first arch     1  1  1  1  0  0  0  0e 1  0  1  0  1s 1  0  0
Oe with beginning of last        1  1  1  0  0  0  1  1  0  1  0  0  1  0
arch 1s.  0e...1s becomes        1  1  1  0  0  0e 1  1  0  1  0  0  1s 0
1...0 in the next line. The         1  1  0  0  1  1  1  0  1  0  0  0
starting 1 and ending 0             1  1  0  0e 1s 1  1  0  1  0  0  0
are removed in the next line           1  0  1  0  1  1  0  1  0  0
reducing number of arches.             1  0e 1  0  1s 1  0  1  0
by one.                                   1  1  0  0  1  0  1  0             cm(2)
                                          1  1  0  0e 1  0  1s 0
                                             1  0  1  1  0  0
                                             1  0e 1s 1  0  0
                                                1  0  1  0                   cm(3)
  Example: T(4,3) 4 starting top arches with 3 closed meanders in history.
		

Crossrefs

Cf. A005315 (row sums), A001181, A005316, A000682.

Formula

T(n,n) = A001181(n).

T(n,2) = A005316(2*n-4)*2 for n > 1.