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.

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, 50, 1312, 1466, 268, 44, 8, 2, 1, 65, 2948, 6812, 1726, 268, 44, 8, 2, 1, 82, 5945, 26048, 11062, 1732, 268, 44, 8, 2, 1, 101, 11026, 84149, 64548, 11617, 1732, 268, 44, 8, 2, 1

Offset: 1

Author

Peter Luschny, Mar 20 2023

Keywords

• nonn
• tabl

Comments

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.

The diagonals, starting from the main diagonal, converge to A141147?

Examples

			Triangle T(n, k) starts:
  [ 1]   1;
  [ 2]   2,     1;
  [ 3]   5,     2,     1;
  [ 4]  10,     8,     2,     1;
  [ 5]  17,    40,     8,     2,     1;
  [ 6]  26,   161,    44,     8,     2,    1;
  [ 7]  37,   506,   263,    44,     8,    2,   1;
  [ 8]  50,  1312,  1466,   268,    44,    8,   2,  1;
  [ 9]  65,  2948,  6812,  1726,   268,   44,   8,  2, 1;
  [10]  82,  5945, 26048, 11062,  1732,  268,  44,  8, 2, 1;
  [11] 101, 11026, 84149, 64548, 11617, 1732, 268, 44, 8, 2, 1.
.
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):
110100100101, 111001001001, 111100010001, 111110000001, 111010010010,
111100100100, 111110001000, 111111000000.
		

Links

• Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Enumeration of Dyck paths with air pockets, arXiv:2202.06893 [cs.DM], 2022-2023.
• Peter Luschny, Fibonacci meanders.

Crossrefs

Cf. A361574 (row sums), A202409, A141147.

Programs

SageMath

# using functions 'isMeander' and 'isFibonacci' from A361574.
def FibonacciMeandersByLeftTurns(m: int, n: int) -> list[int]:
    size = m * n; A = [0] * n; k = -1
    for a in range(0, size + 1, m):
        S = [i < a for i in range(size)]
        for c in Permutations(S):
            if c[0] == 0: break
            if not isFibonacci(c): continue
            if not isMeander(m, c): continue
            A[k] += 1
        k += 1
    return A
for n in range(1, 12):
    print(FibonacciMeandersByLeftTurns(3, n))