A361574 -id:A361574 - cp's OEIS frontend

A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.

1, 3, 6, 13, 30, 70, 167, 405, 992, 2450, 6090, 15214, 38165, 96069, 242530, 613811, 1556856, 3956316, 10070871, 25674210, 65541142, 167517654, 428635032, 1097874434, 2814611701, 7221917871, 18544968768, 47655572191, 122544150258, 315313433594, 811792614547

Offset: 1

Peter Luschny, Jan 15 2012

nonn

Empirically the partial sums of A051291. - Sean A. Irvine, Jul 13 2022

The above conjecture was proved by Baril et al., which also give a formal definition of the Fibonacci meanders and describe a bijection with a certain class of peakless grand Motzkin paths of length n. - Peter Luschny, Mar 16 2023

			a(3) = 6 = card({100001, 100100, 110000, 111001, 111100, 111111}).

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.

Cf. A110236, A110198, A202411, A203611, A051291, A358734.

Cf. A361574 (case m=3).

A201631 := n -> add(A202411(k),k=0..2*n-1): seq(A201631(i),i=1..9);
# Alternative, using the g.f. of Baril et al.:
S := (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R)*R):
R := (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2): ser := series(S, x, 33):
seq(coeff(ser, x, n), n = 1..31); # Peter Luschny, Mar 16 2023
# Using a recurrence:
a := proc(n) option remember; if n < 5 then return [0, 1, 3, 6, 13][n + 1] fi;
(n*(2*n - 1)*(2*n - 3)*(n - 5)*a(n - 5) - (n - 4)*(2*n - 1)^2*(3*n - 5)*a(n - 4) + (2*n - 5)*(n - 3)*(2*n^2 - 3*n + 2)*a(n - 3) - (2*n - 3)*(n - 2)*(2*n^2 - 3*n + 5)*a(n - 2) + (3*n - 4)*(2*n - 1)*(2*n - 5)*(n - 1)*a(n - 1))/(n*(2*n - 3)*(2*n - 5)*(n - 1)) end: seq(a(n), n = 1..31); # Peter Luschny, Mar 16 2023

a[n_] := Sum[A202411[k], {k, 0, 2 n - 1}];
Array[a, 31] (* Jean-François Alcover, Jun 29 2019 *)

a(n) = Sum_{k=0..2n-1} A202411(k).

a(n) = [x^n] (x^2 - x + 1 - R)/((x - 1)*(x^2 - x - 1 + R) * R), where R = (((x - 3)*x + 1)*(x^2 + x + 1))^(1/2). (This is Theorem 21 in Baril et al.) - Peter Luschny, Mar 16 2023

A361681 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 3.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Mar 20 2023

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?

			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.

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.

Cf. A361574 (row sums), A202409, A141147.

# 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))

A361575 Number of Fibonacci meanders of length n.

Original entry on oeis.org

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

Peter Luschny, Mar 16 2023

For an overview of the terms and functions used, compare A361574. The corresponding sequence counting meanders without the requirement to be Fibonacci is A199932.

			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.

Peter Luschny, Fibonacci meanders.

Cf. A361574, A199932, A198060.

# 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);

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 2.

Original entry on oeis.org

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

Peter Luschny, Mar 31 2023

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.

			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.

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.

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

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

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A201631 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 2.

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A361681 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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A361575 Number of Fibonacci meanders of length n.

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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.

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A361681 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 3.

A361894 Triangle read by rows. T(n, k) is the number of Fibonacci meanders with a central angle of 360/m degrees that make mk left turns and whose length is mn, where m = 2.