Questions tagged [recurrence-relations]

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

A recurrence relation is an equation that recursively defines a sequence or multidimensional array of values: once one or more initial terms are given, each further term of the sequence or array is defined as a function of the preceding terms.

Simple examples include the geometric sequence $a_{n}=r a_{n-1}$, which has the closed-form $a_{n}=r^n a_0$, the aforementioned Fibonacci sequence with initial conditions $f_0=0,f_1=1$ and recurrence $f_{n+2}=f_{n+1}+f_n$, and series: the sequence $S_n =\sum_{k=1}^{n} a_k$ can be written as $S_n= S_{n-1}+a_n$.

The term order is often used to describe the number of prior terms used to calculate the next one; for instance, the Fibonacci sequence is of order 2.

See the Wikipedia page for more information.

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find recurrence relation $T(n)=2T(n/2) +\log_2(n)$

$$\begin{align*} &T(n) = 2T(n/2) + \log_2(n)\\ &T(1) = 0 \end{align*}$$ $n$ is a power of $2$ solve the recurrence relation my work so far: unrolling this, we have $$\begin{align*} T(n) &= 4T(n/4) + \log_2(n) -1\\ &= 8T(n/8) + 2\log_2(n)…
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How to decide whether a sequence, defined with a quadratic map converges or not?

Given a sequence $(x_n)_{n \in \mathbb{N}^0}$ defined by a quadratic map $$x_{n+1} = ax_n^2 + bx_n + c$$ with $x_0 = 0$ and $a,b,c \in \mathbb{R}$, is there a fast way to decide whether the sequence converges or not? So far I've been able to come…
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How can you solve this recurrence?

How can I solve this recurrence? $$B_{k}=1+\frac{n-k-1}{n} B_{k+1} + \frac{kx}{n},\qquad x>0$$ This is defined for $1 \leq k \leq n-1$ and $n \geq 2$. When $k=n-1$ then we can see that $B_{n-1} = 1+ \dfrac{(n-1)x}{n}$ so this is effectively the…
user54551
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A simpler proof of Recursion Theorem

Recursion Theorem: Let $A$ be a set, $a\in A$, and $f \colon A\to A$ a mapping. Then there exists a unique mapping $g \colon \Bbb N\to A$ such that 1. $g(0)=a$ 2. $g(n+1)=f(g(n))$ I formalize a proof based on the comments of Noah Schweber at here…
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If $y=\mathrm{e}^x\big(a\sin x+b\cos x\big)$, then express $y^{(n)}$ in terms of $y$ and $y'$.

Let $y=e^x(a\sin x+b\cos x)$. Show $y''=py'+ qy$ for some constants $p$ and $q$; and express all higher derivatives as linear combinations of $y'$ and $y$. I got to $y''=2y'-2y$, but I'm not sure how to do the linear combinations part, I don't know…
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Solve $T(n)=T\left(\frac{n}{2}\right)+2^{n}$

Solve $$T(n)=T\left(\frac{n}{2}\right)+2^{n}.$$ My…
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Find a recurrence relation for the number of strings of length $n$ over $\{1,2,3,4,5\}$ such that :

For $n\in\mathbb{N^+}$ we'll denote $a_n$ as the number of strings of length $n$ over $\{1,2,3,4,5\}$ such that the sum of each 2 adjacent characters is not divisible by 3. Find a recurrence relation of the sequence $a_n$. Then, find a closed form…
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$a_n $ is a positive integer for any $n\in \mathbb {N} $.

Let $(a_n)_{n\geq 1}$ be a sequence defined by $a_{n+1}=(2n^2+2n+1)a_n-(n^4+1 )a_{n-1} $. $a_1=1$, $a_2=3$. I have to show that $a_n $ is a positive integer for any $n\in \mathbb {N}, n\geq 1$. I tried to prove it by induction but it doesn't work.
rafa
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Simplify Permutation Inversion Count Recurrence

Let $I_{n, k}$ denote the number of permutations $\pi \in S_n$ with exactly $k$ inversions. Using the convention $I_{n, k} = 0$ for $k < 0$ and $\displaystyle k > \frac{n(n - 1)}{2}$, we first have the well-known sum $\displaystyle I_{n, k} =…
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Julia set of $x_n = \frac{ x_{n-1}^2 - 1}{n}$

Consider the following iterations : $x_0 = z$ Where $z$ is complex. $x_n = \frac{ x_{n-1}^2 - 1}{n}$ It is well known that for real $z > 3$ the sequence grows double exponentially. It is known that for $z = 3$ the sequence grows linear ; in fact…
mick
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Is it possible to prove this version of recursion theorem from the simpler one?

The simpler version: Let $A$ be a set, $a\in A$, and $f \colon A\to A$ a mapping. Then there exists a unique mapping $g \colon \Bbb N\to A$ such that 1. $g(0)=a$ 2. $g(n+1)=f(g(n))$ The version I want to prove: Let $A$ be a set, $a\in A$, and…
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Solving recurrence relation

$A(n) = A(n-1) + B(n-1)$ $B(n) = A(n-1)$ $A(1) = 2\ ,\ B(1) = 1 $ Please help to find closed form of $C(n) = A(n) + B(n)?$
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Does this dynamical system show an "absorbing area" or a "chaotic area"?

I am following the technical report by C.Mira: "Noninvertible maps: notion of chaotic area vs that of strange attractor" in order to characterize the behavior some dynamical systems of my own. In the mentioned article, there is a definition of…
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Logarithms with a Fraction as Base

How does one solve a logarithmic expression where the base is a fraction? In my example I am trying to solve the following: $$ n^{\log_\frac{3}{2}(1)} \tag{1} $$ This is related to using the "master theorem" to solve recurrence relations. People…
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Limit of recurrence sequence

I have to find a limit (or prove it doesn't exist) for the following recurrence sequence. $a_1 = 2; a_{n+1} = \frac{1}{2}(a_n + \frac{2}{a_n})$ Now I know, in order to find the limit, I first need to prove that the sequence is monotonic and bounded.…
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