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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.
0
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
1,3
COMMENTS
The terms of this sequence can also be derived from sequences of consecutively numbered stamps folded with stamp 1 on top.
FORMULA
T(n,n) = A001181(n).
T(n,2) = A005316(2*n-4)*2 for n > 1.
EXAMPLE
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.
Sequence in context: A324253 A208385 A186634 * A139213 A344873 A306079
KEYWORD
nonn,tabl,more
AUTHOR
Roger Ford, Jun 27 2023
STATUS
approved