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A361575
Number of Fibonacci meanders of length n.
0
1, 3, 5, 11, 13, 30, 34, 71, 97, 177, 233, 481, 610, 1157, 1677, 3027, 4181, 8016, 10946, 20379, 29534, 52461, 75025, 140748, 196778, 355979, 526123, 933044, 1346269, 2469992, 3524578, 6342729, 9400985, 16487211
OFFSET
1,2
COMMENTS
For an overview of the terms and functions used, compare A361574. The corresponding sequence counting meanders without the requirement to be Fibonacci is A199932.
EXAMPLE
Fibonacci meanders with length 6 can have the central angle 360/m, where m is in divisors(6) = {1, 2, 3, 6}. In total there are a(6) = 30 such meanders, the list shows their binary representation together with the multiplicity with which they appear.
100000 x 1, 100001 x 2, 100010 x 1, 100100 x 2, 100101 x 1, 101000 x 1,
101001 x 1, 101010 x 1, 110000 x 2, 110001 x 2, 110010 x 1, 110100 x 1,
110101 x 1, 111000 x 2, 111001 x 2, 111010 x 1, 111100 x 2, 111101 x 1,
111110 x 1, 111111 x 4.
MAPLE
# The list A was computed with the functions given in A361574. They correspond to the columns in the table shown in the reference.
A := [[1, 2, 4, 7, 12, 20, 33, 54, 88, 143, 232, 376, 609, 986, 1596, 2583, 4180, 6764, 10945, 17710, 28656, 46367, 75024, 121392, 196417, 317810, 514228, 832039, 1346268, 2178308, 3524577, 5702886, 9227464, 14930351], [1, 3, 6, 13, 30, 70, 167, 405, 992, 2450, 6090, 15214, 38165, 96069, 242530, 613811, 1556856], [1, 3, 8, 21, 68, 242, 861, 3151, 11874, 45192, 173496], [1, 3, 10, 35, 154, 858, 4723, 25625], [1, 3, 12, 61, 360, 3058], [1, 3, 14, 111, 878], [1, 3, 16, 209], [1, 3, 18, 403], [1, 3, 20], [1, 3, 22], [1, 3, 24], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1, 3], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1], [1]];
with(LinearAlgebra): # a(n) is the sum of row n of this table.
row := k -> [seq(`if`(irem(n, k) <> 0, 0, A[k][n/k]), n = 1..34)]:
M := Transpose(Matrix([seq(row(n), n = 1..34)])):
seq(add(m, m = Row(M, n)), n = 1..34);
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Peter Luschny, Mar 16 2023
STATUS
approved