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%I #14 Mar 31 2023 06:52:41
%S 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 50,1312,1466,268,44,8,2,1,65,2948,6812,1726,268,44,8,2,1,82,5945,
%U 26048,11062,1732,268,44,8,2,1,101,11026,84149,64548,11617,1732,268,44,8,2,1
%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 = 3.
%C 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 The diagonals, starting from the main diagonal, converge to A141147?
%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] 5, 2, 1;
%e [ 4] 10, 8, 2, 1;
%e [ 5] 17, 40, 8, 2, 1;
%e [ 6] 26, 161, 44, 8, 2, 1;
%e [ 7] 37, 506, 263, 44, 8, 2, 1;
%e [ 8] 50, 1312, 1466, 268, 44, 8, 2, 1;
%e [ 9] 65, 2948, 6812, 1726, 268, 44, 8, 2, 1;
%e [10] 82, 5945, 26048, 11062, 1732, 268, 44, 8, 2, 1;
%e [11] 101, 11026, 84149, 64548, 11617, 1732, 268, 44, 8, 2, 1.
%e .
%e 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 110100100101, 111001001001, 111100010001, 111110000001, 111010010010,
%e 111100100100, 111110001000, 111111000000.
%o (SageMath) # using functions 'isMeander' and 'isFibonacci' from A361574.
%o def FibonacciMeandersByLeftTurns(m: int, n: int) -> list[int]:
%o size = m * n; A = [0] * n; k = -1
%o for a in range(0, size + 1, m):
%o S = [i < a for i in range(size)]
%o for c in Permutations(S):
%o if c[0] == 0: break
%o if not isFibonacci(c): continue
%o if not isMeander(m, c): continue
%o A[k] += 1
%o k += 1
%o return A
%o for n in range(1, 12):
%o print(FibonacciMeandersByLeftTurns(3, n))
%Y Cf. A361574 (row sums), A202409, A141147.
%K nonn,tabl
%O 1,2
%A _Peter Luschny_, Mar 20 2023
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