[seqfan] Re: 8/5 Sequence

[franktaw at netscape.net]

>    Hi, Seqfan
>
>    8/5 Sequence :
>
>    IF  x=Even  THEN  x=x/2
>    IF  x=Odd  AND
>        8*x=0  Mod 5  THEN  x=(8*x+5)/5
>        8*x=1  Mod 5  THEN  x=(8*x+4)/5
>        8*x=2  Mod 5  THEN  x=(8*x+3)/5
>        8*x=3  Mod 5  THEN  x=(8*x+2)/5
>        8*x=4  Mod 5  THEN  x=(8*x+1)/5

This can be simplified: if x is odd, then next x is floor(8/5*x)+1.

>...
>    It is easy to generalize the sequence.
>
>    m/ n Sequence :
>    IF  x=Even  THEN  x=x/2
>    IF  x=Odd  AND
>        m*x=0    Mod n  THEN  x=(m*x+n)/n
>        m*x=1    Mod n  THEN  x=(m*x+n-1)/n
>        m*x=2    Mod n  THEN  x=(m*x+n-2)/n
>        ………....
>        m*x=n-1  Mod n  THEN  x=(m*x+1)/n

No need for separate m and n; let r be the ratio (m/n), and if x is 
odd, the next x is floor(r*x)+1.  And now we don't even need r to be 
rational.

>...

When r is the golden ratio, (sqrt(5)+1)/2, starting with 3, we get 
A001595, a sequence of all odd numbers, so there is unlimited growth.  
This may shed some light on what is happening with 8/5.

(Is A128587 a signed version of A001595?
http://www.research.att.com/~njas/sequences/?q=id%3AA001595|id%3AA128587 
)

Franklin T. Adam
s-Watters