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How to prove that this sequence converges?

Let the sequence defined recursively by the equation: $$ a_n = a_{a_{n - 1} } + a_{n - a_{n - 1} } $$ How can I prove that $$ \mathop {\lim }\limits_{n \to \infty } \frac{{a_n }} {n} $$

EDIT: $ a_0 = a_1 = 1 $ Thanks

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marked as duplicate by t.b., Chris Eagle, Hans Lundmark, Willie Wong Sep 14 '11 at 14:13

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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The question is incomplete. (1) What are you trying to prove about the limit? (2) Are there any initial conditions on the sequence? (I’m not sure why you’re asking: you’ve not accepted an answer to any of your five previous questions.) –  Brian M. Scott Sep 14 '11 at 10:15
    
@Brian: If you're interested in some context and background on this sequence, look at David Speyer's answer in the link I gave above. –  t.b. Sep 14 '11 at 10:48
    
@Theo: Thanks. Willie’s comment is correct (hence my answer below, though I wasn’t aware of the earlier question at that time). The Kubo/Vakil paper sounds interesting; I’ll have to see if I can dig up a copy. (I just retired, so I’ve lost my electronic access through the university, at least until my emeritus status goes through.) –  Brian M. Scott Sep 14 '11 at 10:58
    
So trivial is the problem? If this is the case, sorry for be so stupid –  August Sep 14 '11 at 12:08
    
I download it , thx –  August Sep 14 '11 at 12:56

1 Answer 1

If you calculate the first few terms of the sequence, you should very easily be able to conjecture a closed form for $a_n$ that works for all $n>0$. Proving the conjecture is an easy exercise in mathematical induction. And once you have that, the value of the limit is obvious.

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