The "commas" sequence revisited

[Eric Angelini]

Hello Math-Fun and SeqFans,

I would like to come back to this, if you don't mind
(this was one year ago):
http://www.groupsrv.com/hobby/about405191.html

Ed Murphy had asked (quote):

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> What if d = abs(a(n+1)+a(n)), i.e. the sequence is not 
necessarily monotonically increasing? Which choices 
for a(1) result in a block? Which choices result in a loop? 

Sample sequence: 
1, 12, 35, 94, 135, 78, 159, 251, 239, 148, 62, 91, 74, 
115, 166, 105, 156, 88, 4, 48, 129, 221, 209, 118, 199,...

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[Me]:
-- Could someone check if the sequence starting with a(1)=1
ever enters a loop (should we always keep the terms as low
as possible -- although we would still avoid blocks)?

-- Is there a seq which could be a loop by itself?
   a-->b-->c-->d-->e-->f-->a

Best,
É.