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