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%I A363910 #24 Dec 30 2023 19:59:36
%S A363910 1,0,2,0,2,6,0,6,14,22,0,28,56,86,92,0,162,298,428,518,422,0,1076,
%T A363910 1868,2562,3096,3144,2074,0,7852,13076,17292,20624,21990,19366,10754
%N 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.
%C A363910 The terms of this sequence can also be derived from sequences of consecutively numbered stamps folded with stamp 1 on top.
%F A363910 T(n,n) = A001181(n).
%F A363910 T(n,2) = A005316(2*n-4)*2 for n > 1.
%e A363910 n\k  1     2      3     4     5     6     7     8
%e A363910 1:   1
%e A363910 2:   0     2
%e A363910 3:   0     2      6
%e A363910 4:   0     6     14    22
%e A363910 5:   0    28     56    86    92
%e A363910 6:   0   162    298   428   518   422
%e A363910 7:   0  1076   1868  2562  3096  3144  2074
%e A363910 8:   0  7852  13076 17292 20624 21990 19366 10754
%e A363910 Closed meander:         Closed meander split with bottom rotated right
%e A363910 4 top arches            to form top of semi-meander with 8 arches
%e A363910     ______                   ______
%e A363910    / ____ \                 / ____ \
%e A363910   / / __ \ \               / / __ \ \              __
%e A363910  / / /  \ \ \             / / /  \ \ \            /  \
%e A363910 / / / /\ \ \ \           / / / /\ \ \ \  /\  /\  / /\ \
%e A363910 \ \/ /  \/  \/           binary representation of semi-meander
%e A363910  \__/                    1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0
%e A363910                     Semi-meander top arches with no covering center arch  =  cm
%e A363910                               START:          center |
%e A363910 Reduction of semi-meander:    1  1  1  1  0  0  0  0  1  0  1  0  1  1  0  0 cm(1)
%e A363910 Combine end of first arch     1  1  1  1  0  0  0  0e 1  0  1  0  1s 1  0  0
%e A363910 Oe with beginning of last        1  1  1  0  0  0  1  1  0  1  0  0  1  0
%e A363910 arch 1s.  0e...1s becomes        1  1  1  0  0  0e 1  1  0  1  0  0  1s 0
%e A363910 1...0 in the next line. The         1  1  0  0  1  1  1  0  1  0  0  0
%e A363910 starting 1 and ending 0             1  1  0  0e 1s 1  1  0  1  0  0  0
%e A363910 are removed in the next line           1  0  1  0  1  1  0  1  0  0
%e A363910 reducing number of arches.             1  0e 1  0  1s 1  0  1  0
%e A363910 by one.                                   1  1  0  0  1  0  1  0             cm(2)
%e A363910                                           1  1  0  0e 1  0  1s 0
%e A363910                                              1  0  1  1  0  0
%e A363910                                              1  0e 1s 1  0  0
%e A363910                                                 1  0  1  0                   cm(3)
%e A363910   Example: T(4,3) 4 starting top arches with 3 closed meanders in history.
%Y A363910 Cf. A005315 (row sums), A001181, A005316, A000682.
%K A363910 nonn,tabl,more
%O A363910 1,3
%A A363910 _Roger Ford_, Jun 27 2023

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