Hofstadter-like sequences

[Eric Angelini]

Hello SeqFans,
I was wondering: do sequences which have a different start than A005228 converge to said A005228?
 
Here is A005228 (first diff. below):
 
1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213
  2  4  5   6   8   9  10  11  13  14  15  16   17   19   20   21   22
 
If we start with "2" (instead of "1") the new seq converges immediately:
 
2, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213
  1  4  5   6   8   9  10  11  13  14  15  16   17   19   20   21   22
 
But what about starting with "3":
 
3, 4, 6, 11, 18, 26, 35, 45, 57, 70, 84, 99, 115, 132, 151, 171, ... ?
  1  2  5   7   8   9  10  12  13  14  15  16   17   19   20   21
 
If someone knows a general paper on those "Hofstadter-like" sequences, 
I would be pleased to read more -- I'm not sure there is something like
that here : 
http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst <http://www.research.att.com/~njas/sequences/Sindx_Ho.html#Hofst> 
 
Best,
E.
 
 
 
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