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Given a sequence $u_n$ such that

$u_1 = 1$

$u_{2n} = n + u_n$

$u_{2n+1} = n^2 + u_nu_{n+1}$

How to solve for closed-form of $u_n$? I really don't know where to start.

Cranky
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    One simple way to start is to tabulate $u_n$ for small values of $n$; I did it for $n=1,\ldots,16$ and got $$1,2,3,4,10,6,21,8,56,15,85,12,162,28,217,16;,$$ which is not very promising as far as a closed form is concerned. And it’s more than enough to discover that OEIS has no entry for the sequence. (It is clear, though, that $u_{2^n}=2^n$ for all $n$, for whatever that may be worth.) – Brian M. Scott Feb 13 '15 at 21:31
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    And somewhat more generally, $u_{2^m k} = (2^m - 1) k + u(k)$ for odd $k$. – Robert Israel Feb 13 '15 at 22:26

1 Answers1

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Why do you think there is a closed form? It's unlikely for an arbitrary nonlinear recurrence. Neither OEIS nor gfun come up with anything.

Robert Israel
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