[seqfan] a(n) --> a(n)' = 3 x string(a) x string(b)

Eric Angelini Eric.Angelini at kntv.be
Mon Oct 26 19:39:11 CET 2009

Hello SeqFans,
(more examples here:
 http://www.cetteadressecomportecinquantesignes.com/3xx.htm )

Let's see what the two rules hereunder (and the split technique) produce:

a) if a(n) has one single digit, then a(n)-->a(n)'= [3×a(n)]; example: 7 ->[3×7] = 21;
b) if a(n) has more than one digit then we split a(n) into two strings s1 and s2 such that a(n)-->a(n)'= [3×s1×s2] 

How to split:

- for a two-digit number [t,u] we have s1=t and s2=u (example: 28 splits into 2 and 8)
- for a three, four, five...-digit number we take all possible ways of partitioning a(n) into two strings (not allowing string s2 to have a leading zero). Example for a(n)=123405:

123405 --> s1=1 and s2=23405
    or --> s1=12 and s2=3405
    or --> s1=123 and s2=405
    or --> s1=12340 and s2=5
    We do not accept the split s1=1234 and s2=05

So, if a(n)= 123405 we have a(n)'= 3×1×23405 =  70215
                         or a(n)'= 3×12×3405 = 122580
                         or a(n)'= 3×123×405 = 149445
                         or a(n)'= 3×12340×5 = 185100

We then make a(n)'=a(n) and iterate the process.


A few (interesting?) questions arise about fixed points, pseudo-loops,
flight lenghts, etc.


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