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<DIV><FONT face="MS UI Gothic"> Hello, Seqfans.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> I considered about an analog
of 3x+1 sequence in Gaussian Integer.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> [Definition of 3x+1
sequence]</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> a(n)=(3*a(n-1)+1)/2^k ,
where 2^k is the highest power of two dividing 3*a(n-1)+1
. <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> [A translation to Gaussian
integer] <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic">
3 -> 2+i or 1+2i
.... Secondly small
prime<BR> 1 ->
1
....
Unit<BR> 2 ->
1+i
.... The smallest
prime</DIV>
<DIV><FONT size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> (2+i)x+1 sequence is defined
as follows<BR></DIV></FONT></FONT>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> a(n)=((2+i)*a(n-1)+1)/(1+i)^k
, where (1+i)^k is the highest power of (1+i) dividing (2+i)*a(n-1)+1
. <BR> <BR></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> (1+2i)x+1 sequence is defined
as follows<BR>
<DIV><FONT face="MS UI Gothic" size=2></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> a(n)=((1+2i)*a(n-1)+1)/(1+i)^k
, where (1+i)^k is the highest power of (1+i) dividing (1+2i)*a(n-1)+1
. <BR></DIV></FONT>
<DIV> </DIV>
<DIV> [Examples]</FONT></DIV></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> (2+i)x+1 sequence
:</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> S_1 1, 1+2i, 1, 1+2i,
....<BR> S_2 3, 2+5i, 3, 2+5i,
....<BR> S_3 7, 4+11i, 6+7i, 3+10i, ....
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> (1+2i)x+1 sequence :
</FONT></DIV>
<DIV><FONT face="MS UI Gothic"> T_1 1, 1, 1, 1,
.... </FONT></DIV>
<DIV><FONT face="MS UI Gothic"> T_2 3, 2+3i,
2+5i, <FONT> 1+8i, 6+i, .... </FONT></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> T_3 5, 3+10i, 1, 1,
....</FONT></DIV>
<DIV><FONT face="MS UI Gothic"><FONT size=2></FONT> </DIV>
<DIV><FONT size=2></FONT><FONT size=2></FONT><BR> Numbers are
calculated in the first quadrant of Z[i] plane.</FONT></DIV>
<DIV><FONT face="MS UI Gothic"></FONT> </DIV>
<DIV><FONT face="MS UI Gothic"> I calculated only six
examples on real number line by
hand.<BR> I am not sure if they are
collect, because the factorization is difficult without a
computer.
.<BR> Do S_3 , T_2 become
periodic?<BR> </FONT><FONT face="MS UI Gothic"></FONT></DIV>
<DIV><FONT face="MS UI Gothic"> Yasutoshi
<BR> </FONT></DIV></BODY></HTML>