 The Syracuse conjecture for U(0)={1,2,3,4,...,128} -monodimensional display of the parities- [La conjecture de Syracuse pour U(0)={1,2,3,4,...,128} -visualisation monodimensionnelle des parités-].

• The Syracuse sequence is defined as follows:

## ``` U = N (an integer number) > 0 0 if U is even : n U n U = ---- n+1 2 else : 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.

• The horizontal and vertical axes represent the integer numbers {1, 2, 3, 4,...}. Each vertical line (with abscissa equals to N) displays the parity of each U(n) starting at U(0)=N and the color of each of its points {N,U(n)} is function of the parity of 'U(n)' (Red=even, White=odd). For example with N=7, the X=7 vertical line is displayed by means of the 17 following points:
```                    {7,7}  P=1 (White)
{7,22} P=0 (Red)
{7,11} P=1 (White)
{7,34} P=0 (Red)
{7,17} P=1 (White)
{7,52} P=0 (Red)
{7,26} P=0 (Red)
{7,13} P=1 (White)
{7,40} P=0 (Red)
{7,20} P=0 (Red)
{7,10} P=0 (Red)
{7,5}  P=1 (White)
{7,16} P=0 (Red)
{7,8}  P=0 (Red)
{7,4}  P=0 (Red)
{7,2}  P=0 (Red)
{7,1}  P=1 (White)
```
where 'P' denotes the parity.

One can concatenate all the parities {P} giving the following binary number:
```                    10000100010010101
```
that is 67733 as a decimal number.

Here are the parities -as decimal and then as binary numbers- of 128 different sequences starting from U(0)=1 to U(0)=128.

See some related visualizations (including this one):                          (CMAP28 WWW site: this page was created on 04/18/2019 and last updated on 04/23/2019 11:24:35 -CEST-)

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