A201631 -id:A201631 - cp's OEIS frontend

A361574 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 3.

1, 3, 8, 21, 68, 242, 861, 3151, 11874, 45192, 173496, 673042

Offset: 1

Peter Luschny, Mar 16 2023

A binary curve C is a pair (m, S) such that
   (a) S is a list with values in {L, R} that
   (b) starts with an L, and
   (c) m > 0 is an integer that divides the length of S.
Given a binary curve C = (m, S), we call m the modulus of the curve and S the steps of C. 'L' stands for a positively oriented arc (left turn) and 'R' for a negatively oriented arc (right turn).
Let SS denote a pair of consecutive steps in C. The direction d of a curve at n starts with d = 0, and for n > 0, d = d + 1 if SS = LL and d = d - 1 if SS = RR; otherwise, d remains unchanged.
A binary curve C = (m, S) is a meander if the values d mod m are assumed with equal frequency. A meander is a closed smooth curve in the plane, possibly self-overlapping (see the plots).
A binary curve 'changes direction' if two consecutive steps are of the same type, i.e., is a pair of steps of the form LL or RR.
A Fibonacci meander is a meander that does not change direction to the left except at the beginning of the curve, where any number of left turns can appear.

			For n = 4 the Fibonacci meanders with central angle 120 degrees and length 12, written as binary strings (L = 1, R = 0), are:
100000010001, 100010000001, 110000000001, 100000100100, 100100000100, 100010001000,
110000001000, 100100100000, 110001000000, 111000000000, 110100100101, 111001001001,
111100010001, 111110000001, 111010010010, 111100100100, 111110001000, 111111000000,
111111110001, 111111111000, 111111111111.

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.
Susanne Wienand, Counting meanders by dynamic programming.
Susanne Wienand, Example of a meander, Plot and animation.

Cf. A000071 (m=1), A201631 (m=2), A197657.

def isFibonacci(S: list[bool]) -> bool:
    dir, os = True, S[0]
    for s in S:
        if s and os:
            if not dir:
                return False
            dir = True
        else:
            dir = False
        os = s
    return True
def isMeander(n: int, S: list[bool]) -> bool:
    if S[0] != True:
        return False
    vec = [0] * n
    max = len(S) // n
    dir, ob = 0, True
    for b in S:
        if b and ob:
            dir += 1
        elif not b and not ob:
            dir -= 1
        dir = dir % n
        v = vec[dir] = vec[dir] + 1
        if v > max:
            return False
        ob = b
    return True
def FibonacciMeanders(m: int, steps: int) -> int:
    size = m * steps
    count = 0
    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
            count += 1
        print(count)
    return count
def FibonacciMeandersColumn(m: int, size: int) -> list[int]:
    return [FibonacciMeanders(m, n) for n in range(1, size + 1)]
print([FibonacciMeandersColumn(3, n) for n in range(1, 7)])

A110198 Antidiagonal sums of number triangle A110197.

Original entry on oeis.org

1, 2, 4, 9, 20, 46, 109, 262, 638, 1569, 3886, 9680, 24225, 60856, 153368, 387573, 981742, 2491934, 6336721, 16139616, 41166912, 105139773, 268841100, 688157430, 1763206441, 4521749642, 11605580290, 29809644693, 76621733444, 197074591420, 507193044993

Offset: 0

Paul Barry, Jul 15 2005

Partial sums of A051286.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A051286, A110197, A201631.

CoefficientList[Series[1/((1-x)*Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

G.f.: 1/((1-x)*sqrt((1+x+x^2)*(1-3x+x^2))); a(n) = sum{k=0..floor(n/2), sum{i=0..n-2k, binomial(i+k, k)^2}}.
a(n) = sum{i=0..2n, A202411(i)}. - Peter Luschny, Jan 16 2012
Conjecture: n*a(n) +(-3*n+1)*a(n-1) +n*a(n-2) +(-n+2)*a(n-3) +(3*n-5)*a(n-4) +(-n+2)*a(n-5)=0. - R. J. Mathar, Nov 15 2012
a(n) ~ sqrt(100+45*sqrt(5)) * ((sqrt(5)+3)/2)^n / (10*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 08 2014
Equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

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))

cp's OEIS Frontend

A361574 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 3.

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A110198 Antidiagonal sums of number triangle A110197.

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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 mk left turns and whose length is mn, where m = 2.

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cp's OEIS Frontend

A361574 a(n) is the number of Fibonacci meanders of length m*n and central angle 360/m degrees where m = 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A110198 Antidiagonal sums of number triangle A110197.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

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.