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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Evaluating $\sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}}$

Inspired by Ramanujan's problem and solution of $\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \ldots}}}$, I decided to attempt evaluating the infinite radical $$ \sqrt{1 + \sqrt{2 + \sqrt{4 + \sqrt{8 + \ldots}}}} $$ Taking a cue from Ramanujan's solution method,…
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Divisibility property for sequence $a_{n+2}=-2(n-1)(n+3)a_n-(2n+3)a_{n+1}$

Let $(a_n)$ be the sequence uniquely defined by $a_1=0,a_2=1$ and $$ a_{n+2}=-2(n-1)(n+3)a_n-(2n+3)a_{n+1} $$ Can anybody show (or provide a counterexample) that $p|a_{p-2}$ and $p|a_{p-1}$ for any prime $p\geq 5$ ? I have checked this fact for…
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I have a problem understanding the proof of Rencontres numbers (Derangements)

I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation $$D_{n,0}=\left[\frac{n!}{e}\right]$$ where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation…
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How to prove that the Binet formula gives the terms of the Fibonacci Sequence?

This formula provides the $n$th term in the Fibonacci Sequence, and is defined using the recurrence formula: $u_n = u_{n − 1} + u_{n − 2}$, for $n > 1$, where $u_0 = 0$ and $u_1 = 1$. Show that $$u_n = \frac{(1 + \sqrt{5})^n - (1 -…
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Does this sequence always terminate or enter a cycle?

I've been fiddling with the recursive sequence defined as follows: $$\begin{equation} f_n=\begin{cases} a, & n=1.\\ b, & n=2.\\ c, & n=3.\\ f_{n-1}f_{n-2}f_{n-3} \mod[f_{n-1}+f_{n-2}+f_{n-3}], & n>3. \end{cases} \end{equation}$$ And no…
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Is $\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin x\cdots\right)\right)=\frac4{\pi}\sum\limits_{k=0}^\infty\frac{\sin(2k+1)x}{2k+1}$?

We can see intuitively that $$ f(x)=\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\sin\left(\frac{\pi}{2}\cdots\sin{x}\cdots\right)\right)\right) $$ is the square wave with period $2\pi$ and has the value $0$ at the jumps,…
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Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns out to be periodic with period 9, i.e., $$ a_n =…
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show this sequence always is rational number

let $\{a_{n}\}$ such $a_{1}=-8$,and such $$4\sqrt[3]{a_{n}}+5\sqrt[3]{a_{n+1}}=3\sqrt[3]{7(a_{n}+1)(a_{n+1}+1)}$$ show that $$a_{n}\in Q,\forall n\in N^{+}$$ I try let $a_{2}=x$,and for $n=1$, then we…
math110
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Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following conjecture true? A conjecture : If $a_n$ is an integer, then $n\le 8$. I conjectured this by using computer, but I don't have any…
mathlove
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Show that $\lim\limits_{n\to\infty}\frac{a_1+a_2+\dots+a_n}{n^2}$ exists and is independent of the choice of $a$

Suppose $f:\mathbb{R}\to\mathbb{R}$ has period 1, and for some $q\in(0,1)$: $$|f(x)-f(y)|\leq q|x-y|\quad \forall x,y$$ Let $g(x)=x+f(x)$, for any $a\in\mathbb{R}$,define the following sequence: $$a_1=a,\quad a_2=g(a_1),\quad a_3=g(a_2),\quad…
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Numbers of distinct values obtained by inserting $+ - \times \div ()$ in $\underbrace{2\quad2 \quad2 \quad2\quad...\quad 2}_{n \text{ times}}$

This question is inspired by How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis? (By the way, I sincerely hope this kind of questions can receive more attention) Insert $+ - \times \div ()$ in $$\underbrace{2\quad2 \quad2…
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A surprising result about the product of Blaschke matrices

I have verified analytically the conjecture described bellow up to $n=4$, but have had no success trying to prove it. Any help would be much appreciated. Setup Let $\{\lambda_i\}_{i=1}^n$ be real numbers and $g_i:\mathbb R \to \mathbb R$ for all…
mzp
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Number of ways to partition a rectangle into n sub-rectangles

How many ways can a rectangle be partitioned by either vertical or horizontal lines into n sub-rectangles? At first I thought it would be: f(n) = 4f(n-1) - 2f(n-2) where f(0) = 1 and f(1) = 1 but the recurrence relation only counts the…
puri
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Identifying this chaotic (?) recurrence relation

Update: I'm working on an interactive bifurcation diagram: http://matt-diamond.com/sineMap.html Here's the image when the starting coordinates are [0.5, 0.5] The bifurcation diagrams differ depending on whether or not the coordinates are equal to…
Matt D
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Why is this family of dynamical systems able to produce spirals and clusters of points?

I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a complex discrete map as follows: (Disclaimer, tl:dr:…