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I know that this sequence is found in some papers etc, but nowhere is this little problem solved, only discuss it as a trivial, at least I could not do it, so I ask for help. Let the following sequence defined recursively: $$ \eqalign{ & a_1 = a_2 = 1 \cr & a_n = a_{a_{n - 1} } + a_{n - a_{n - 1} } \cr} $$ prove that the subsequence $ a_{2^k} $ is such that $$ a_{2^k } = 2^{k - 1} $$

EDITED: I only changed the initial values replacing $a_0 $ by $a_2$ because it does not hold in the other case, now yes

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