# Tagged Questions

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

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### Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$

I am trying to work out a problem on some old analysis qual exam, and managed to reduce it to this question, but I can't seem to figure out this final step: Consider the sequence defined by the ...
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### Links between difference and differential equations?

Does there exist any correspondence between difference equations and differential equations? In particular, can one cast some classes of ODEs into difference equations or vice versa?
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### Family of GCDs all equal to $2$

Is it true that $$\gcd\left(5^{2^n} + 1, 13^{2^n} + 1\right) = 2$$ for all $n \in \mathbf{Z}_{\geq 0}$? I'm continually stumped with this and verifying it numerically is quite expensive very ...
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### Linear homogeneous recurrence relations with repeated roots; motivation behind looking for solutions of the form $nx^n$?

If we have a linear homogeneous recurrence relation, such as $t_{k+1}=4t_k-4t_{k-1}$, and attempt to find solutions of the form $t_n=x^n$ for some $x \in \mathbb{R} \setminus \{0\}$, we obtain the ...
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### Show that $2^n(\cos^n(\frac{2\pi}{9})+\cos^n(\frac{4\pi}{9})+\cos^n(\frac{8\pi}{9}))\in\mathbb{Z}$

$a_n=2^n\left[\cos^n\left(\dfrac{2\pi}{9}\right)+\cos^n\left(\dfrac{4\pi}{9}\right)+\cos^n\left(\dfrac{8\pi}{9}\right)\right]$. Show that $a_n\in\mathbb{Z}$ for all $n\in\mathbb{Z}$. Find the last ...
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### Prove that this particular sequence contains an infinite number of sixes

Given the sequence $$2,7,1,4,7,4,2,8,\ldots$$ which begins with $2, 7$ and is constructed by multiplying successive pairs of its members and adjoining the results as the next one or two members of ...
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### Find asymptotic of recurrence sequence

Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$. The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$. ...
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### What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?

This looks like a question asked earlier, but it isn't. $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad & \text{ if } n>1 \\ 1 & \text{ if }n=1\end{cases}$$ My professor gave this to me in ...
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### How many permutations of a multiset have a run of length k?

Background $\newcommand\ms[1]{\mathsf #1}\def\msP{\ms P}\def\msS{\ms S}\def\mfS{\mathfrak S}$Suppose I have $n$ marbles of $c$ colors, where $c≤n$. Let $n_i$ denote the number of marbles of color $i$....
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### Closed form for $a_{n+1} = (a_n)^2+\frac{1}{4}$

I've been given the following sequence: \begin{align*} &a_0 = 0; \\ &a_{n+1} = (a_n)^2+\frac{1}{4}. \end{align*} I also have to prove that whatever I come up with is correct, but that will ...
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### Solving randomized recurrence relation

I'm looking at the random sequence $x_n$ with $x_0=x_1=1$ and $$x_{n+1}=2x_n\pm x_{n-1}$$ where we choose the $\pm$ sign independently with equal probability. Now ...
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### Recurrence for the partition numbers

I'm reading Analytic Combinatorics [PDF] book by Flajolet and Sedgewick, and I can't figure out one of the steps in the derivation of the $P_n$ — number of partitions of size $n$ (or coefficients in ...
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### A polynomial sequence

I have a sequence of polynomials $Q_k(x, y)$, $k\geq 1$ defined recursively as follows: $Q_1=x$. There is a sequence of polynomials $p_j(y)$ of degree $j$ such that $Q_{2m}$ is of the form \begin{...
### What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$?
Let $f_0=2$ and $f_n=f_{n-1}+\ln(f_{n-1})$. What is a good asymptotic to the sequence $f_n$? With good I mean much better than $f_n \sim \dfrac{3n \ln(2)\ln(n)}{2}$.
Consider some arbitrary number sequence like the decimal expansion of $\pi$ = {3, 1, 4, 1, 5, 9, 2}. Prepend the sequence with the number $1$ so that you get {1, 3, 1, 4, 1, 5, 9, 2}. Then plug it ...