The Syracuse conjecture for U(0)={5,6,7,8,...,68} -polar coordinates display- [*La conjecture de Syracuse pour U(0)={5,6,7,8,...,68} -visualisation en coordonnées polaires-*].

- 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 sixty-four different sequences from U(0)=5 (lower left)
to U(0)=68 (upper right). For each sequence U(n) the following "star" is generated:
Rho(n) = U (with a renormalization inside [0,1]) n

2.pi Theta(n) = ------.n nM+1

X(n) = Rho(n).cos(Theta(n)) Y(n) = Rho(n).sin(Theta(n))

where 'nM' denotes the first 'n' such as:U = 1 nM

The colors used are a function of 'n' (from Dark Blue [n=0] to White with an increasing luminance ).

See some related visualizations (including this one):

(CMAP28 WWW site: this page was created on 01/21/2013 and last updated on 02/28/2022 11:11:41 -CET-)

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