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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.

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

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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