A005316 -id:A005316 - cp's OEIS frontend

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

Roger Ford, Jun 27 2023

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

			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.

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

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

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

A378944 Triangle read by rows: T(n,k) = number of stamp foldings with stamp #1 first, n stamps and stamp #2 covered by exactly one fold. k = the stamp number before the fold covering stamp #2 divided by 2. See examples.

Original entry on oeis.org

2, 4, 8, 6, 20, 12, 48, 24, 28, 132, 60, 56, 348, 144, 112, 162, 1008, 396, 280, 324, 2812, 1044, 672, 648, 1076, 8420, 3024, 1848, 1620, 2152

Offset: 5

Roger Ford, Dec 11 2024

The conjectured formula for the numbers in T(n,k) involves two unsolved sequences, semi-meanders and meandric numbers.

			                          _____    __         ______________
Vertical lines = stamp#  |     |  |  |       |   __    __   |   __
Horizontal lines = folds 1  5  2  3  4       |  |  |  |  |  |  |  |
                            |  |__|  |       1  6  5  4  3  2  8  7
                            |________|          |  |__|  |__|     |
     fold 4-5 covers stamp #2  k = 4/2          |_________________|
                        Example: T(5,2)      fold 6-7 covers stamp #2 k = 6/2
                                                               Example: T(8,3)
Irregular triangle begins:
   n\k  (2)  (3)  (4)  (5)  (6)
    5:   2
    6:   4
    7:   8    6
    8:   20   12
    9:   48   24   28
   10:   132  60   56
   11:   348  144  112  162
   12:   1008 396  280  324
   13:   2812 1044 672  648  1076
   14:   8420 3024 1848 1620 2152

Cf. A000682, A005316, A077054, A301620.

T(n,k) = 2 * A000682(n+1-2*k) * A077054(k-1).

cp's OEIS Frontend

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.

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