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A361681
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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?
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LINKS
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EXAMPLE
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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.
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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.
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PROG
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(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))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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