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

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6
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1answer
496 views

Solving a simple recurrence relation

I have the following recurrence relation: $a_0=1$ $a_{n}=pa_{n+1}+qa_{n-1}$ Where $p+q=1$. This relation arises in analyzing a "gambler's ruin" situation. It is claimed that the general solution ...
6
votes
2answers
414 views

Convergence/Divergence of a particular infinite nested radical

Is it known if the following infinite nested radical converges or diverges (for $n \in \mathbb N$)?: $$R(n) = \sqrt{n+\sqrt{(n+1)+\sqrt{(n+2)+ \cdots}}}$$ I recently became interested in these ...
6
votes
4answers
496 views

How to find limit of a sequence defined by recurrence formula

I have the following problem: Let $a_{n}$ be the recurrence $$a_{n+1}=a_{n}+2a_{n-1}$$ with $a_{0}=0$ and $a_{1}=1$. Can you help me find $$\lim_{n\to\infty} \frac{a_{n+1}}{a_{n}}$$ for $n\geq ...
6
votes
1answer
3k views

Find a big-O estimate for $f(n)=2f(\sqrt{n})+\log n$

While self-studying Discrete Mathematics, I found the following question in the book "Discrete Mathematics and Its Applications" from Rosen: Suppose the function $f$ satisfies the recurrence ...
6
votes
2answers
830 views

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 ...
6
votes
2answers
109 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
6
votes
1answer
300 views

Recurrence relation telescoping

Hi there I am trying to solve the following recurrence relation using telescoping. How would I go about doing it? $$T(n) = \frac 2n \Big(T(0) + T(1) + \ldots+ T(n-1)\Big) + 5n$$ Assuming $n\ge 1$
6
votes
1answer
103 views

How to prove that this recursively defined sequence converges to $e$?

Let $a_1=0,a_2=1,$ and $a_{n+2}=\dfrac{(n+2) a_{n+1}-a_n}{n+1}$. Prove that $\lim_{n\to \infty}a_n=e$. I know that $\lim_{n\to\infty}\left(1+\frac1{2!}+\frac1{3!}+...+\frac1{n!}\right)=e$ and $a_n = ...
6
votes
2answers
132 views

Generating Functions: Solving a Second-Order Recurrence

I'm self-studying generating functions (using GeneratingFunctionology as a text). I came across this programming problem, which I immediately recognized as a modification of the Fibonacci sequence. ...
6
votes
1answer
320 views

Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
6
votes
1answer
124 views

Limit related to the recursion $a_{n+1}=(1-n^{-1/4}a_n)a_n$

Let $0<a_{1}<1$, with $$a_{n+1}=a_{n}(1-n^{-\frac{1}{4}}a_{n})$$ Does there a real numbers $A$ which make the limit $$ ...
6
votes
2answers
142 views

The limit of a recurrence relation (with resistors)

Background to problem (not too important): My proposed solution: The infinitely long element, , however complex, can be represented as a single resistor of resistance $R$. Remembering the ...
6
votes
1answer
238 views

Finding ($2012$th term of the sequence) $\pmod {2012}$

Let $a_n$ be a sequence given by formula: $a_1=1\\a_2=2012\\a_{n+2}=a_{n+1}+a_{n}$ find the value: $a_{2012}\pmod{2012}$ So, in fact, we have to find the value of ...
6
votes
2answers
85 views

Is the sequence $x_{n+1} = \frac{x_n + \alpha}{x_n + 1}$ convergent?

Fix $\alpha > 1$, and consider the sequence $(x_n)_n \geq 0$ defined by $x_0 > \sqrt \alpha$ and $$x_{n+1} = \frac{x_n + \alpha}{x_n + 1}, n = 0, 1, 2, \dots$$ Does this sequence converge, ...
6
votes
1answer
130 views

How find $a_{n}$ if $a_{n+1}=\sqrt{2a_n+1}$

let $a_{1}=\dfrac{\sqrt{2}}{4}$ and such $$a_{n+1}=\sqrt{2a_{n}+1}$$ find $a_{n}$ my idea:let $a_{n}=\dfrac{1}{2}\cos{x_{n}}$ $$\Longrightarrow ...
6
votes
1answer
184 views

Line in a proof on p69 in Cassel's Local Fields

I'm trying to read the proof of LEMMA 6.1 (Nagell) Let $u_n$ be defined by $u_0=0$, $u_1=1$ and $u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$. Then $u_n=\pm1$ only for $n=1,2,3, ...
6
votes
1answer
1k views

Repertoire Method Clarification Required ( Concrete Mathematics )

In the book Concrete Mathematics, chapter 2, section 2.2 -- sums and recurrences, page 26 (2nd edition), the authors talk about the following example: Given the general recurrence $$ R(0) = \alpha ...
6
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1answer
263 views

What is the asymptotic behavior of a linear recurrence relation (equiv: rational g.f.)?

The question sounds simple: find the roots of the characteristic equation, take the one with the largest absolute value, and find its coefficient. Repeated roots do not substantially complicate ...
6
votes
1answer
257 views

Common terms in general Fibonacci sequences

Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
6
votes
2answers
362 views

How to prove $\lim_{n\to\infty}\frac{\log_{2}{(f_{n})}}{(\log_{2}{n})^2}=\frac{1}{2}$?

For any $n\in N$, such $f_{1}=1$, and such $$f_{2n+1}=f_{2n}=f_{2n-1}+f_{n},$$ prove that $$\lim_{n\to\infty}\dfrac{\log_{2}{(f_{n})}}{(\log_{2}{n})^2}=\dfrac{1}{2}.$$
6
votes
1answer
133 views

Recurrence Relation

How do I solve: $k(k+1)a_{k}=2(\lambda k-1)a_{k-1}+(a-\lambda^2)a_{k-2}$ where $\lambda$ and $a$ are constants, and similar other recurrence relations?
6
votes
2answers
324 views

unorthodox solution of a special case of the master theorem

I am asking for references regarding a special case of the master theorem. This theorem seems to appear quite a lot on this site, prompting me to study it in more detail, e.g. see my posts here and ...
6
votes
1answer
150 views

Prove that every element of $a_{n+2013}=\frac{a_{n+1}a_{n+2}…a_{n+2012}+1}{a_n}$ is an integer

Given $\displaystyle a_1=a_2=\cdots=a_{2013}=1$ and $\displaystyle a_{n+2013}=\frac{a_{n+1}a_{n+2}\cdots a_{n+2012}+1}{a_n}$. Prove that $a_{n+2013}\in\mathbb{N}$ for all $n\in\mathbb{N}$. I ...
6
votes
0answers
84 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
6
votes
0answers
40 views

Recurrence relations with multiple roots of auxiliary equation

Suppose we have a homogeneous linear recurrence relation of the form $$u_{n+r}+a_1u_{n+r-1}+\dots +a_ru_n=0$$ The auxiliary equation is then $$f(x)=x^r+a_1x^{r-1}+\dots +a_r=0$$ It is well known that ...
6
votes
0answers
155 views

Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
5
votes
4answers
304 views

General expression of $f(a, b)$ if $f(a, b)=f(a-1,b) + f(a, b-1) + f(a-1, b-1)$?

$f(a,b) = f(a-1, b) + f(a-1, b-1) + f(a, b-1), ab \neq 0$ $f(a,b) = 1, ab = 0$ So what is $f(a, b)$?
5
votes
4answers
404 views

What explains this bizarre behavior?

Consider the sequence $x_{n+1} = 2x_{n}-\frac{1}{x_n},n\geq0 $. For $x_{0} = 0,87$ we have $$ \begin{aligned} X(1) &\approx 0,590574712643678 \\ X(2) &\approx -0,512116436915835\\ X(3) ...
5
votes
2answers
181 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and maybe easy question, but I was not able to find an answer. If $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a linear change of coordinates it can be brought to the ...
5
votes
3answers
358 views

How to solve this recurrence using generating functions?

Exercise: For $n \geq 0$ let $a_n = \sum \limits_{i=0}^n (i^2- 2i + 1)$ a) Show that $$a_{n+4} -4a_{n+3} + 6a_{n+2} - 4a_{n+1} + a_n = 0, n \geq 0$$ b) Identify the genereating series ...
5
votes
1answer
185 views

How to solve this recurrence relation $a_{n+2} = \sqrt{a_{n+1}\cdot a_{n}}$?

I am trying to solve the recurrence: $$ a_{n+2} = \sqrt{a_{n+1}\cdot a_{n}} $$ but here is a problem for me. After few steps I have this: $$ a_n^2 = a_{n-1}\cdot a_{n-2} $$ and I don't now what to do ...
5
votes
3answers
742 views

Analysis of Algorithms: Solving Recursion equations-$T(n)=3T(n-2)+9$

I need your help with solving this recursion equation: $T(n)=3T(n-2)+9$. with the initial condition : $T(1)=T(2)=1$. I need to find $T(n)$, the complexity of the algorithm which works that way. I ...
5
votes
2answers
264 views

A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of ...
5
votes
5answers
723 views

Solving a set of recurrence relations

I have the 7 following reccurence relations: $A_n = B_{n-1} + C_{n-1}$ $B_n = A_n + C_{n-1}$ $C_n = B_n + C_{n-1}$ $D_n = E_{n-1} + G_{n-1}$ $E_n = D_n + F_{n-1}$ $F_n = G_n + C_n$ $G_n = E_n + ...
5
votes
5answers
308 views

What particular solution should I guess for this relation?

Just trying to solve a non-homogeneous recurrence relation: $$f(n) = 2f(n-1) + n2^n$$ $$f(0) = 3$$ Characteristic equation is: $$f(n) - 2f(n-1) = 0$$ $$a-2 = 0$$ $$a = 2$$ Homogeneous ...
5
votes
1answer
151 views

Solve a recurrence relation $D_{n} = nD_{n-1} + (n-1)!$

As the title states, I need a solution of this recurrence but I'll provide my own solution and ask if there is an easier, simpler solution using some deeper knowledge about recurrence relations. ...
5
votes
3answers
132 views

Prove that this recurrence always cycles

If $n$ is a nonnegative integer, let $S_n=\{0, 1, 2, \dots, 2n+1\}$. For $t\in S_n$ repeatedly perform if t is even t = t/2 else t = (n + 1 + ⌊t/2⌋) ...
5
votes
3answers
83 views

sequence $U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$

Is there a general formula for $U_k$ defined by $$U_k=a_0U_{k-1}+a_1U_{k-2}+\cdots+a_{k-1}U_0$$ where the $a_i$ are in arithmetic progression and $U_0=1$? Do there always exist $c,d$ such that ...
5
votes
2answers
102 views

Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
5
votes
1answer
157 views

Strange Recurrence: What is it asymptotic to?

So I have the following recurrence relation for the growth rate of an algorithm: $T(n)$ = time taken by algorithm to solve problem of size n: $$T(n) = T(n-1) + T(\lceil(n/2)\rceil)$$ Clearly this ...
5
votes
2answers
202 views

Combinatorial explanation to following recurrence relation $a_n = 2 a_{n-1} + a_{n-2}$

Question was the following: $a_n$ is the number of ternary strings (strings of 0,1,2) which contain no consecutive zeros and no consecutive ones. Find a formula for $a_n$? By brute force, I found a ...
5
votes
4answers
182 views

Finding the general term of two related recurrence relations

I'm trying to find the general term of the recurrence relations $\quad a_{n+1}=a_n+\text hb_n$ $\quad b_{n+1}=b_n-\text ha_n $ $\quad a_0=0, \quad b_0=1$ I tried finding the terms, ...
5
votes
3answers
925 views

How to solve this recurrence relation

What are some ways to solve this recurrence relation: $a(n+1)=2 a(n) - a(n-1) -1, \text{ with }a(0)=0, a(10)=0?$ I tried to first convert this inhomogeneous equation into a homogeneous one following ...
5
votes
3answers
129 views

The number of length-n ternary sequences with even ones and even zeroes

Just starting to appreciate recurrence relations Let $T_n = $ number of length-n ternary sequences with an even number of ones and an even number of zeroes. $T_0 = 1$, because $0$ is an even number, ...
5
votes
2answers
117 views

Why should we suspect that the recurrence $T(n) = T(n-1) + n(n-1)$ satisfies a polynomial identity?

In the question Algorithms: Recurence Relation, the author asked about the recurrence relation $$T(n) = T(n-1) + n(n-1)$$ and one of the answers proposed assuming $T(n)$ is polynomial, then ...
5
votes
1answer
168 views

Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$

I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $ then the general equation looks like the following: $$ F_n = F_{n-1}(F_{n-2}) + F_{n-3} $$ So far I have got the ...
5
votes
4answers
127 views

Recurrence equation: $u_n = 4u_{n−1} + 4u_{n−2}$ ; is $4x+4 = 4$ the characteristic equation?

Given this recurrence equation: $u_1 = 0, u_2 = 1$ $u_n = 4u_{n−1} + 4u_{n−2}$ Is the correct characteristic equation: $4x+4 = 4$ EDIT: Complete solve: The characteristic equation is ...
5
votes
2answers
366 views

Series of sequence converges?

Given the recursively defined sequence $$ a_2 = 2(C+1)a_0 $$ and $$ a_{n+2}=\frac{\left(n(n+6)+4(C+1)\right)a_n - 8(n-2)a_{n-2}}{(n+1)(n+2)}. $$ that I got from the Frobenius method applied to an ODE, ...
5
votes
2answers
154 views

Solving a recurrence for a probability?

I came across the following recurrence relation when exploring properties of a certain type of randomized perfect binary tree: $$ T(0) = \frac{1}{2} $$ $$ T(k + 1) = 1 - T(k)^2 $$ (Specifically, ...
5
votes
1answer
185 views

Finding the general formula $a_n$ for $a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$

How to calculate the general formula $a_n$ for the following sequence: $$a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$$ where $a_1=\frac{1}{2}, a_2=\frac{1}{4}$