# 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. The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 (= 0 / 1 ) and 1 (= 1 / 1 ), by taking successive mediants.

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 :

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

• next term = child
• Previous terms = parents

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

### n

take the Misiurewicz point for Z^n+c and increase n ( propososed by Owen Maresh)

The constants for the quadratic, cubic, and quartic are:

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

## 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.
• the number is a period of the miget ( main cardioid of the midget) that appear the first time in that order
${\begin{array}{cccccccc}3&5&7&9&11&\ldots &(2n+1)\cdot 2^{0}&\ldots \\3\cdot 2&5\cdot 2&7\cdot 2&9\cdot 2&11\cdot 2&\ldots &(2n+1)\cdot 2^{1}&\ldots \\3\cdot 2^{2}&5\cdot 2^{2}&7\cdot 2^{2}&9\cdot 2^{2}&11\cdot 2^{2}&\ldots &(2n+1)\cdot 2^{2}&\ldots \\3\cdot 2^{3}&5\cdot 2^{3}&7\cdot 2^{3}&9\cdot 2^{3}&11\cdot 2^{3}&\ldots &(2n+1)\cdot 2^{3}&\ldots \\&\vdots \\\ldots &2^{n}&\ldots &2^{4}&2^{3}&2^{2}&2&1\end{array}}$

"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."

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

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

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

## sequence of fraction in the elephant valley

In the elephant valley ( 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 of $2^{p}$ . One can easly create a numeric value that is too large to be represented within the available storage space ( integer overflow ). For example $2^{34}$  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

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

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

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) $t={\frac {p}{q}}$  (double) t $c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}$
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
• virtual Siegel disk
• ? Leau-Fatou flower ?

### 1 over 3

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

n t $c_{t}={\frac {2e^{2\pi it}-e^{4\pi it}}{4}}$
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 )

Approximations to the reciprocal golden ratio by finite continued fractions, or ratios of Fibonacci numbers
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).