A361894 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.

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, 186, 70, 20, 6, 2, 1, 9, 176, 462, 246, 70, 20, 6, 2, 1, 10, 261, 1016, 812, 252, 70, 20, 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

Offset: 1

Author

Peter Luschny, Mar 31 2023

Keywords

• nonn
• tabl

Comments

For an overview of the terms used see A361574. A201631 gives the row sums of this triangle.

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).

The diagonals, starting from the main diagonal, apparently converge to A000984.

Examples

			Triangle T(n, k) starts:
  [ 1]  1;
  [ 2]  2,   1;
  [ 3]  3,   2,    1;
  [ 4]  4,   6,    2,    1;
  [ 5]  5,  16,    6,    2,    1;
  [ 6]  6,  35,   20,    6,    2,   1;
  [ 7]  7,  66,   65,   20,    6,   2,   1;
  [ 8]  8, 112,  186,   70,   20,   6,   2,  1;
  [ 9]  9, 176,  462,  246,   70,  20,   6,  2,  1;
  [10] 10, 261, 1016,  812,  252,  70,  20,  6,  2, 1;
  [11] 11, 370, 2025, 2416,  917, 252,  70, 20,  6, 2, 1;
  [12] 12, 506, 3730, 6435, 3256, 924, 252, 70, 20, 6, 2, 1.
.
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):
k = 1: 11000000, 10010000, 10000100, 10000001;
k = 2: 11110000, 11100100, 11100001, 11010010, 11001001, 10100101;
k = 3: 11111100, 11111001;
k = 4: 11111111.
		

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. A201631 (row sums), A361681 (m=3), A132812, A361574, A103371, A000984.

Programs

SageMath

# using function 'FibonacciMeandersByLeftTurns' from A361681.
for n in range(1, 12):
    print(FibonacciMeandersByLeftTurns(2, n))