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A361681 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 = 3.

%I A361681 #14 Mar 31 2023 06:52:41
%S A361681 1,2,1,5,2,1,10,8,2,1,17,40,8,2,1,26,161,44,8,2,1,37,506,263,44,8,2,1,
%T A361681 50,1312,1466,268,44,8,2,1,65,2948,6812,1726,268,44,8,2,1,82,5945,
%U A361681 26048,11062,1732,268,44,8,2,1,101,11026,84149,64548,11617,1732,268,44,8,2,1
%N A361681 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 = 3.
%C A361681 For an overview of the terms used see A361574, which gives the row sums of this triangle. The corresponding sequence counting meanders without the requirement of being Fibonacci is A202409.
%C A361681 The diagonals, starting from the main diagonal, converge to A141147?
%H A361681 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 A361681 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FibonacciMeanders">Fibonacci meanders</a>.
%e A361681 Triangle T(n, k) starts:
%e A361681   [ 1]   1;
%e A361681   [ 2]   2,     1;
%e A361681   [ 3]   5,     2,     1;
%e A361681   [ 4]  10,     8,     2,     1;
%e A361681   [ 5]  17,    40,     8,     2,     1;
%e A361681   [ 6]  26,   161,    44,     8,     2,    1;
%e A361681   [ 7]  37,   506,   263,    44,     8,    2,   1;
%e A361681   [ 8]  50,  1312,  1466,   268,    44,    8,   2,  1;
%e A361681   [ 9]  65,  2948,  6812,  1726,   268,   44,   8,  2, 1;
%e A361681   [10]  82,  5945, 26048, 11062,  1732,  268,  44,  8, 2, 1;
%e A361681   [11] 101, 11026, 84149, 64548, 11617, 1732, 268, 44, 8, 2, 1.
%e A361681 .
%e A361681 T(4, 2) = 8 counts the Fibonacci meanders with central angle 120 degrees and length 12 that make 6 left turns. Written as binary strings (L = 1, R = 0):
%e A361681 110100100101, 111001001001, 111100010001, 111110000001, 111010010010,
%e A361681 111100100100, 111110001000, 111111000000.
%o A361681 (SageMath) # using functions 'isMeander' and 'isFibonacci' from A361574.
%o A361681 def FibonacciMeandersByLeftTurns(m: int, n: int) -> list[int]:
%o A361681     size = m * n; A = [0] * n; k = -1
%o A361681     for a in range(0, size + 1, m):
%o A361681         S = [i < a for i in range(size)]
%o A361681         for c in Permutations(S):
%o A361681             if c[0] == 0: break
%o A361681             if not isFibonacci(c): continue
%o A361681             if not isMeander(m, c): continue
%o A361681             A[k] += 1
%o A361681         k += 1
%o A361681     return A
%o A361681 for n in range(1, 12):
%o A361681     print(FibonacciMeandersByLeftTurns(3, n))
%Y A361681 Cf. A361574 (row sums), A202409, A141147.
%K A361681 nonn,tabl
%O A361681 1,2
%A A361681 _Peter Luschny_, Mar 20 2023