The Syracuse conjecture for U(0)={5,6,7,8,...,68} -'pseudo-ramdom walk' display: 'The Syracuse Pulsar'- [La conjecture de Syracuse pour U(0)={5,6,7,8,...,68} -visualisation sous forme d'une 'pseudo-marche aléatoire': 'Le Pulsar de Syracuse'-].

• The Syracuse sequence is defined as follows:

## ``` U = N (an integer number [un nombre entier]) > 0 0 if U is even [si U est pair] : n n U n U = ---- n+1 2 else [sinon] : U = 3*U + 1 n+1 n ```

• The Syracuse conjecture states that sooner or later the {[[4,] 2,] 1} sequence will appear whatever the starting number N (and then repeats itself obviously ad vitam aeternam). For example with U(0)=7:
```                    U(0)  =  7
U(1)  = 22
U(2)  = 11
U(3)  = 34
U(4)  = 17
U(5)  = 52
U(6)  = 26
U(7)  = 13
U(8)  = 40
U(9)  = 20
U(10) = 10
U(11) =  5
U(12) = 16
U(13) =  8
U(14) =  4
U(15) =  2
U(16) =  1
```

Here are 256 different sequences starting from U(0)=1 to U(0)=256.

• This picture is a circular display of 64 different sequences from U(0)=5 (on the right-hand side) to U(0)=68 -rotating counterclockwise-. For each sequence {U(n)} a "pseudo-ramdom walk" is generated: the parities {0,1} of each U(n) define two angles {-THETA,+THETA} at the center. The 64 starting U(0) values are located at the periphery of the structure -dark orange, with a radius proportional to the length of the sequence {U(n)}-, when the 64 {4,2,1} sequences are at the center -white-.

See some related visualizations (including this one):

(CMAP28 WWW site: this page was created on 10/28/2021 and last updated on 02/28/2022 11:11:54 -CET-)

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