# Tagged Questions

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### Help with a recurrence with even and odd terms [duplicate]

I have the following recurrence that I've been pounding on: $$a(0)=1\\ a(1)=1\\ a(2)=2\\ a(2n)=a(n)+a(n+1)+n\ \ \ (\forall n>1)\\ a(2n+1)=a(n)+a(n-1)+1\ \ \ (\forall n\ge 1)$$ I don't have much ...
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### How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
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### Closed form for a non-linear recurrence

Does equation $a_n=\sqrt{a_{n-1}+6}$ with $a_1=6$ have a closed form? I've found no linearization method. Any suggestion or hint will be highly appreciated.
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### Proof of clockwise towers of Hanoi variant recursive solution

This is from one of the exercises in "Concrete Mathematics", and is something I'm doing privately, not homework. This is a variant on the classic towers of Hanoi, where all moves must be made ...
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### recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
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### Common term for differential equations and recurrence relations

Recently I have been working with recurrence relations (mostly linear), and systems of coupled recurrence relations. I have noticed a lot of common ground with differential equations. In a way, you ...
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### How can I find an explicit expression for this recursively defined sequence?

We define the sequence $(u_n)_{n=1}^\infty$by:$$u_{n+1}=1+\frac{1}{u_n}$$ How can I find the limit of this sequence as it goes to infinity? By induction, I can prove that it is bounded above and ...
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### Is there any explicit formula for $x_n$?

Let $a$ be a positive real number and $(x_n)$ be the sequence given by $x_1>0,$ $$x_{n+1}=\dfrac{1}{2}\Big(x_n+\dfrac{a}{x_n}\Big).$$ We can prove that $x_n\to\sqrt{a}$ as $n\to\infty.$ My ...
I derived a recursion $$B_n=\frac{1}{n}B_0+\frac{1}{n}\sum_{k=1}^{n-1}\left[\binom{n}{k}-(-1)^{n-k}\binom{n}{k-1}\right]B_k\tag{*}$$ which I know should be satisfied by the moments of the unit ...