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There is an interesting sequence in problem $88$ of Chapter $12.1$ of Stewart's PreCalculus:

$$a_n=a_{n-a_{n-1}}+a_{n-a_{n-2}}$$

with $a_1=a_2=1$. It is defined recursively but contrary to most examples I know where $a_n$ is defined using the $r$ previous terms $a_{n-1}$, $\dots$, $a_{n-r}$ previous terms ($r$ fixed), here the terms used to defined $a_n$ varies. It was hard to believe at first that the sequence is well-defined.

Is there a name for this kind of sequence?

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That is the Hofstadter Q sequence, one of the Hofstadter sequences described by Douglas Richard Hofstadter in his book “Gödel, Escher, Bach.” These are described by non-linear recurrence relations.

The Hofstadter Q sequence is also listed as A005185 in On-Line Encyclopedia of Integer Sequences®.

According to the Wikipedia article, it is unknown if the sequence is well-defined for all $n$.

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