OFFSET
1,2
COMMENTS
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.
EXAMPLE
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.
PROG
(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))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 20 2023
STATUS
approved