# Fractals/Mathematics/sequences

## types of sequences

### Fraction sequences

#### Farey sequence

The Farey sequence of order n is the sequence of completely reduced vulgar fractions between 0 and 1 which when in lowest terms have denominators less than or equal to n, arranged in order of increasing size.

Each Farey sequence starts with the value 0, denoted by the fraction 01, and ends with the value 1, denoted by the fraction 11 (although some authors omit these terms).

Farey Addition = the mediant of two fractions :

 ${\displaystyle {\frac {a}{c}}\oplus {\frac {b}{d}}={\frac {a+b}{c+d}}}$


Terms

• next term = child
• Previous terms = parents[1]

Farey tree = Farey sequence as a tree

Sorted
 F1 = {0/1,                                                                                                          1/1}
F2 = {0/1,                                                   1/2,                                                   1/1}
F3 = {0/1,                               1/3,                1/2,                2/3,                               1/1}
F4 = {0/1,                     1/4,      1/3,                1/2,                2/3,      3/4,                     1/1}
F5 = {0/1,                1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5,                1/1}
F6 = {0/1,           1/6, 1/5, 1/4,      1/3,      2/5,      1/2,      3/5,      2/3,      3/4, 4/5, 5/6,           1/1}
F7 = {0/1,      1/7, 1/6, 1/5, 1/4, 2/7, 1/3,      2/5, 3/7, 1/2, 4/7, 3/5,      2/3, 5/7, 3/4, 4/5, 5/6, 6/7,      1/1}
F8 = {0/1, 1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8, 1/1}


# Sequences and orders on the parameter plane

## Sequences of Misiurewicz points

• external ray for angle 1/(4*2^n) land on the tip of the first branch: 1/4, 1/8, 1/16, 1/32, 1/64, ...
• 1/(6*2^n) - land on the second branch
• principal Misiurewicz point of wake p/q
• primary_separators

### degree

take the Misiurewicz point for ${\displaystyle z^{n}+c}$ and increase n ( proposed by Owen Maresh)

The constant (parameter c) for the quadratic (n=2) , cubic ( n=3), and quartic (n=4) polynomials are:

• (-0.7432918908524301520519705530861564778806 ,0.1312405523087976002753516038253522297699);
• -0.0649150006787816892861875745218343125883 , 0.76821968591243610206311097043854440463 );
• (-0.593611822136354943067129147813253628530 ,0.5405019391915187246754930586066158919613 );

Point c is a Misiurewicz point ${\displaystyle c=M_{23,2}=-0.743291890852430+0.131240552308798i}$

• tip of the longest branch ( ftip )
• The angle 8388607/25165824 or 01010101010101010101010p01 has preperiod = 23 and period = 2
• from wake 12/25 , with
• center c = -0.739829393511579 +0.125072144080321 i and period = 25
• root point c = -0.738203140939397 +0.124839088573366 i

m-describe 53 30 500 -0.7432918908524301 0.1312405523087976 4
the input point was -7.4329189085243008e-01 + 1.3124055230879761e-01 i
nearby hyperbolic components to the input point:

- a period 1 cardioid
with nucleus at 0.00000e+00 + 0.00000e+00 i
the component has size 1.00000e+00 and is pointing west
the atom domain has size 0.00000e+00
the atom domain coordinates of the input point are -nan + -nan i
the atom domain coordinates in polar form are -nan to the east
the atom coordinates of the input point are -0.74329 + 0.13124 i
the atom coordinates in polar form are 0.75479 to the west
the nucleus is 7.54789e-01 to the east of the input point

- a period 2 circle
with nucleus at -1.00000e+00 + 0.00000e+00 i
the component has size 5.00000e-01 and is pointing west
the atom domain has size 1.00000e+00
the atom domain coordinates of the input point are 0.25671 + 0.13124 i
the atom domain coordinates in polar form are 0.28831 to the east-north-east
the atom coordinates of the input point are 0.51342 + 0.26248 i
the atom coordinates in polar form are 0.57662 to the east-north-east
the nucleus is 2.88311e-01 to the west-south-west of the input point
external angles of this component are:
.(01)
.(10)
the point escaped with dwell 472.09881

nearby Misiurewicz points to the input point:

- 24p4
with center at -7.43291890852430202931624325972515e-01 + 1.31240552308797604770845906581477e-01 i
the Misiurewicz domain has size 1.07586e-03
the Misiurewicz domain coordinate radius is 1.1395e-13
the center is 1.21387e-16 to the west of the input point
the multiplier has radius 1.329970173958942893e+00 and angle 0.150434052944417735 (in turns)


The angle  8388607/25165824  or  01010101010101010101010p01 has  preperiod = 23  and  period = 2.
The corresponding parameter ray lands at a Misiurewicz point of preperiod 23 and period dividing 2.
Do you want to draw the ray and to shift c to the landing point?
c = -0.743291890852430                          +0.131240552308798 i    period = 0


## Myrberg-Feigenbaum point

Examples:

• sequence of root points for periods ${\displaystyle p=1*2^{n}}$ ( period doubling cascade) and the limit point of the sequence is the Myrberg-Feigenbaum point ${\displaystyle \mathbf {MF} _{1/2}=c_{0}=-1.401155189093314712...}$
• sequence of root points for periods ${\displaystyle p=1*3^{n}}$ and the limit point of the sequence is the Myrberg-Feigenbaum point ${\displaystyle \mathbf {MF} =c_{0}=-1.7864402555636388747...}$

## Sharkovsky ordering

• It is the infinite sequence of positive integers ( natural numbers)
• It starts from 3 and ends in 1
• It contains infinitely many subsequences.[2]
• the number is a period of the miget ( main pseudocardioid of the midget) that appear the first time in that order
${\displaystyle {\begin{array}{cccccccc}3&5&7&9&11&\ldots &(2n+1)\cdot 2^{0}&\ldots \\3\cdot 2&5\cdot 2&7\cdot 2&9\cdot 2&11\cdot 2&\ldots &(2n+1)\cdot 2^{1}&\ldots \\3\cdot 2^{2}&5\cdot 2^{2}&7\cdot 2^{2}&9\cdot 2^{2}&11\cdot 2^{2}&\ldots &(2n+1)\cdot 2^{2}&\ldots \\3\cdot 2^{3}&5\cdot 2^{3}&7\cdot 2^{3}&9\cdot 2^{3}&11\cdot 2^{3}&\ldots &(2n+1)\cdot 2^{3}&\ldots \\&\vdots \\\ldots &2^{n}&\ldots &2^{4}&2^{3}&2^{2}&2&1\end{array}}}$

"The Sharkovski ordering :

• begins with the odd numbers >= 3 in increasing order ( n is increasing from left to right ),
• then twice these numbers,
• then 4 times them,
• then 8 times them,
• etc.,
• ending with the powers of 2 in decreasing order, ending with 2^0 = 1."[3]

${\displaystyle (2n+1)\cdot 2^{0}\prec (2n+1)\cdot 2^{1}\prec (2n+1)\cdot 2^{2}\prec \ldots \prec 2^{n}}$

It is related with structure of the real slice of the Mandelbrot set ( along real exis):

• chaotic region, which consist of chaotic bands ${\displaystyle B_{m}=(2n+1)\cdot 2^{m}}$:
• ${\displaystyle B_{0}=(2n+1)\cdot 2^{0}}$
• ${\displaystyle B_{1}=(2n+1)\cdot 2^{1}}$
• ${\displaystyle B_{2}=(2n+1)\cdot 2^{3}}$
• ${\displaystyle \ldots }$
• ${\displaystyle B_{\infty }}$
• MF = Myrberg-Feigenbaum point
• periodic region P with period doubling cascade = 2^n
${\displaystyle B\prec MF\prec P}$
${\displaystyle B_{0}\prec B_{1}\prec ...\prec MF\prec ...\prec 2^{1}\prec 2^{0}}$

## sequence of fraction in the elephant valley

In the elephant valley[4][5] ( from parameter plane ) there is a sequence of componts with period p : from 1/2 to 1/p

Note that :

• internal ray 0/1 = 1/1
• internal angle 1/p means that ray goes from period 1 component ( parent period = 1) to period p component ( child period = p)
• as child period grows computations are harder
• exponential growth[6] of ${\displaystyle 2^{p}}$. One can easly create a numeric value that is too large to be represented within the available storage space ( integer overflow[7] ). For example ${\displaystyle 2^{34}}$ is to big for short ( 16 bit ) and long ( 32 bit) integer.

The upper principal sequence of rotational number around the main cardioid of Mandelbrot set[8]

n rotation number = 1/n parameter c
2 1/2 -0.75
3 1/3 0.64951905283833*i-0.125
4 1/4 0.5*i+0.25
5 1/5 0.32858194507446*i+0.35676274578121
6 1/6 0.21650635094611*i+0.375
7 1/7 0.14718376318856*i+0.36737513441845
8 1/8 0.10355339059327*i+0.35355339059327
9 1/9 0.075191866590218*i+0.33961017714276
10 1/10 0.056128497072448*i+0.32725424859374

See :

## sequence of parabolic points on the boundary of main cardioid

${\displaystyle t=\sum _{k\mathop {=} 1}^{n}{\frac {3}{10^{k}}}}$

Here:

• t = internal angle ( or rotation number) of main cardioid
• q = number of the critical orbit (star) arms. It means that one have to do q iterations around fixed point to move one point toward fixed point along arm.
• c is a root point between hyperbolic components of period 1 ( = main cardioid) and period q. This point is at the end ( radius = 1) of internal ray for angle t
k = log10(q) ${\displaystyle t={\frac {p}{q}}}$ (double) t ${\displaystyle c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}}$
1 3/10 0.3 +0.047745751406263+0.622474571220695 i
2 33/100 0.33 -0.106920138306109 +0.649235321397436 i
3 333/1000 0.333 -0.123186752260805 +0.649516204880454 i
4 3333/10000 0.3333 -0.124818625550005 +0.649519024348384 i
5 33333/100000 0.33333 -0.124981862061192 +0.649519052553419 i
6 333333/1000000 0.333333 -0.124998186201184 +0.649519052835480 i
7 3333333/10000000 0.3333333 -0.124999818620069 +0.649519052838300 i
8 33333333/100000000 0.33333333 -0.124999981862006 +0.649519052838329 i
9 333333333/1000000000 0.333333333 -0.124999998186201 +0.649519052838329 i
10 3333333333/10000000000 0.3333333333 -0.124999999818620 +0.649519052838329 i

## sequence from Siegel disk to Leau-Fatou flower

• plain Siegel disk
• digitated Siegel disk[9]
• virtual Siegel disk
• ? Leau-Fatou flower ?

### 1 over 3

${\displaystyle t=[0;3,10^{n},g]=0+{\cfrac {1}{3+{\cfrac {1}{10^{n}+{\cfrac {1}{g}}}}}}}$

n t ${\displaystyle c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}}$
0 0.2763932022500210 +0.1538380639536641 + 0.5745454151066985 i
1 0.3231874668087892 -0.0703924965263780 + 0.6469145331346999 i
2 0.3322326933513446 -0.1190170769366243 + 0.6494880316361160 i
3 0.3332223278292314 -0.1243960357918422 + 0.6495187369145560 i
4 0.3333222232791965 -0.1249395463818515 + 0.6495190496732967 i
5 0.3333322222327929 -0.1249939540657307 + 0.6495190528066729 i
6 0.3333332222223279 -0.1249993954008480 + 0.6495190528380124 i
7 0.3333333222222233 -0.1249999395400276 + 0.6495190528383258 i
8 0.3333333322222222 -0.1249999939540022 + 0.6495190528383290 i
9 0.3333333332222223 -0.1249999993954002 + 0.6495190528383290 i
10 0.3333333333222222 -0.1249999999395400 + 0.6495190528383290 i
11 0.3333333333322222 -0.1249999999939540 + 0.6495190528383290 i

### sequence of fractions tending to the golden mean ( Golden Ratio Conjugate )

n =   1 ;  p_n/q_n =  1.0000000000000000000 =                     1 /                    1
n =   2 ;  p_n/q_n =  0.5000000000000000000 =                     1 /                    2
n =   3 ;  p_n/q_n =  0.6666666666666666667 =                     2 /                    3
n =   4 ;  p_n/q_n =  0.6000000000000000000 =                     3 /                    5
n =   5 ;  p_n/q_n =  0.6250000000000000000 =                     5 /                    8
n =   6 ;  p_n/q_n =  0.6153846153846153846 =                     8 /                   13
n =   7 ;  p_n/q_n =  0.6190476190476190476 =                    13 /                   21
n =   8 ;  p_n/q_n =  0.6176470588235294118 =                    21 /                   34
n =   9 ;  p_n/q_n =  0.6181818181818181818 =                    34 /                   55
n =  10 ;  p_n/q_n =  0.6179775280898876404 =                    55 /                   89
n =  11 ;  p_n/q_n =  0.6180555555555555556 =                    89 /                  144
n =  12 ;  p_n/q_n =  0.6180257510729613734 =                   144 /                  233
n =  13 ;  p_n/q_n =  0.6180371352785145888 =                   233 /                  377
n =  14 ;  p_n/q_n =  0.6180327868852459016 =                   377 /                  610
n =  15 ;  p_n/q_n =  0.6180344478216818642 =                   610 /                  987
n =  16 ;  p_n/q_n =  0.6180338134001252348 =                   987 /                 1597
n =  17 ;  p_n/q_n =  0.6180340557275541796 =                  1597 /                 2584
n =  18 ;  p_n/q_n =  0.6180339631667065295 =                  2584 /                 4181
n =  19 ;  p_n/q_n =  0.6180339985218033999 =                  4181 /                 6765
n =  20 ;  p_n/q_n =  0.6180339850173579390 =                  6765 /                10946
n =  21 ;  p_n/q_n =  0.6180339901755970865 =                 10946 /                17711
n =  22 ;  p_n/q_n =  0.6180339882053250515 =                 17711 /                28657
n =  23 ;  p_n/q_n =  0.6180339889579020014 =                 28657 /                46368
n =  24 ;  p_n/q_n =  0.6180339886704431856 =                 46368 /                75025
n =  25 ;  p_n/q_n =  0.6180339887802426829 =                 75025 /               121393
n =  26 ;  p_n/q_n =  0.6180339887383030068 =                121393 /               196418
n =  27 ;  p_n/q_n =  0.6180339887543225376 =                196418 /               317811
n =  28 ;  p_n/q_n =  0.6180339887482036214 =                317811 /               514229
n =  29 ;  p_n/q_n =  0.6180339887505408394 =                514229 /               832040
n =  30 ;  p_n/q_n =  0.6180339887496481015 =                832040 /              1346269
n =  31 ;  p_n/q_n =  0.6180339887499890970 =               1346269 /              2178309
n =  32 ;  p_n/q_n =  0.6180339887498588484 =               2178309 /              3524578
n =  33 ;  p_n/q_n =  0.6180339887499085989 =               3524578 /              5702887
n =  34 ;  p_n/q_n =  0.6180339887498895959 =               5702887 /              9227465
n =  35 ;  p_n/q_n =  0.6180339887498968544 =               9227465 /             14930352
n =  36 ;  p_n/q_n =  0.6180339887498940819 =              14930352 /             24157817
n =  37 ;  p_n/q_n =  0.6180339887498951409 =              24157817 /             39088169
n =  38 ;  p_n/q_n =  0.6180339887498947364 =              39088169 /             63245986
n =  39 ;  p_n/q_n =  0.6180339887498948909 =              63245986 /            102334155
n =  40 ;  p_n/q_n =  0.6180339887498948319 =             102334155 /            165580141
n =  41 ;  p_n/q_n =  0.6180339887498948544 =             165580141 /            267914296
n =  42 ;  p_n/q_n =  0.6180339887498948458 =             267914296 /            433494437
n =  43 ;  p_n/q_n =  0.6180339887498948491 =             433494437 /            701408733
n =  44 ;  p_n/q_n =  0.6180339887498948479 =             701408733 /           1134903170
n =  45 ;  p_n/q_n =  0.6180339887498948483 =            1134903170 /           1836311903


This is a sequence of rational numbers ( Julia sets are parabolic). It's limit is an irrational number ( Julia set has a Siegel disk).