The Syracuse conjecture for U(0)={5,6,7,8,...,20} -tridimensional display- [*La conjecture de Syracuse pour U(0)={5,6,7,8,...,20} -visualisation tridimensionnelle-*].

- 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 tridimensional display of sixteen different sequences from U(0)=5 (lower left)
to U(0)=20 (upper right). For each sequence U(n) the following tridimensional
set of segments is generated:
X coordinates = {U(0),U(1),U(2),...,U(n),...} Y coordinates = {U(1),U(2),U(3),...,U(n+1),...} Z coordinates = {U(2),U(3),U(4),...,U(n+2),...}

with a renormalization inside [0,1] for the X and Y coordinates when a renormalization inside [-1/2,+1/2] for the Z coordinates is used. 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 01/18/2013 and last updated on 05/06/2020 10:58:19 -CEST-)

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