A361681 - OEIS

%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