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%I #9 Mar 31 2023 07:00:24
%S 1,2,1,3,2,1,4,6,2,1,5,16,6,2,1,6,35,20,6,2,1,7,66,65,20,6,2,1,8,112,
%T 186,70,20,6,2,1,9,176,462,246,70,20,6,2,1,10,261,1016,812,252,70,20,
%U 6,2,1,11,370,2025,2416,917,252,70,20,6,2,1,12,506,3730,6435,3256,924,252,70,20,6,2,1
%N Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make m*k left turns and whose length is m*n, where m = 2.
%C For an overview of the terms used see A361574. A201631 gives the row sums of this triangle.
%C The corresponding sequence counting meanders without the requirement of being Fibonacci is A103371 (for which in turn A103327 is a termwise majorant counting permutations of the same type).
%C The diagonals, starting from the main diagonal, apparently converge to A000984.
%H Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/2202.06893">Enumeration of Dyck paths with air pockets</a>, arXiv:2202.06893 [cs.DM], 2022-2023.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FibonacciMeanders">Fibonacci meanders</a>.
%e Triangle T(n, k) starts:
%e [ 1] 1;
%e [ 2] 2, 1;
%e [ 3] 3, 2, 1;
%e [ 4] 4, 6, 2, 1;
%e [ 5] 5, 16, 6, 2, 1;
%e [ 6] 6, 35, 20, 6, 2, 1;
%e [ 7] 7, 66, 65, 20, 6, 2, 1;
%e [ 8] 8, 112, 186, 70, 20, 6, 2, 1;
%e [ 9] 9, 176, 462, 246, 70, 20, 6, 2, 1;
%e [10] 10, 261, 1016, 812, 252, 70, 20, 6, 2, 1;
%e [11] 11, 370, 2025, 2416, 917, 252, 70, 20, 6, 2, 1;
%e [12] 12, 506, 3730, 6435, 3256, 924, 252, 70, 20, 6, 2, 1.
%e .
%e T(4, k) counts Fibonacci meanders with central angle 180 degrees and length 8 that make k left turns. Written as binary strings (L = 1, R = 0):
%e k = 1: 11000000, 10010000, 10000100, 10000001;
%e k = 2: 11110000, 11100100, 11100001, 11010010, 11001001, 10100101;
%e k = 3: 11111100, 11111001;
%e k = 4: 11111111.
%o (SageMath) # using function 'FibonacciMeandersByLeftTurns' from A361681.
%o for n in range(1, 12):
%o print(FibonacciMeandersByLeftTurns(2, n))
%Y Cf. A201631 (row sums), A361681 (m=3), A132812, A361574, A103371, A000984.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Mar 31 2023
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