# Fractals/Mathematics/sequences
## types of sequences
### Integer sequences

* A137560
* A137867

### 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 $^0$⁄$_1$, and ends with the value 1, denoted by the fraction $^1$⁄$_1$ (although some authors omit these terms).
Farey Addition = the mediant of two fractions :
Terms
* next term = child
* Previous terms = parents
Farey tree = Farey sequence as a tree
See also
* Fibonacci Numbers hidden in the Mandelbrot Set - Numberphile

# 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
** How to compute external angles of principal Misiurewicz point of wake p/q using Devaney's algorithm ?
** program Mandel by Wolf Jung
* primary_separators

### degree

* cubic
take the Misiurewicz point for $z^{n}+c$ 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 $c=M_{23,2}=-0.743291890852430+0.131240552308798i$
** 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

## Myrberg-Feigenbaum point

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

## 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.
* the number is a period of the miget ( main pseudocardioid 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 consist 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 P with period doubling cascade = 2^n
    $B\prec MF\prec P$
    $B_{0}\prec B_{1}\prec ...\prec MF\prec ...\prec 2^{1}\prec 2^{0}$

### Period doubling scenario

* 1/2 family
* Pariod doubling cascade in the Mandelbrot set ( 1/2 family) showed by the exponential mapping
* escape route 1/2
* period doubling

## 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 :
* Slide show of rescaled limbs converging to the Lavaurs elephant - video by Wolf Jung made with Mandel

## 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 2

* Infolding Siegel Disk for c near internal angle t=1/2 on the boundary of main cardioid of Mandelbrot set

### 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 )

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

# More

* orbit
** period of the orbit

# References

* ↑ Finding parents in the Farey tree by Claude Heiland-Allen
* ↑ Sharkovskii's theorem in wikipedia
* ↑ The On-Line Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
* ↑ muency : elephant valley
* ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
* ↑ integer number in wikipedia
* ↑ Integer overflow in wikipedia
* ↑ Mandel Set Combinatorics : Principal Series
* ↑ scholarpedia : Siegel_disks , Quadratic_Siegel_disks, Digitation
* Hardy-DivergentSeries
* Mandelbrot Buds and Branches by Timothy Chase: