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

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7
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2answers
321 views

q-Analogue of the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$.

The stirling numbers of the second kind satisfy the formula $x^n=\sum_k\left\{n\atop k\right\}(x)_k$, where $(x)_k$ is the falling factorial. Consider the $q$-analog recursive definition of the ...
7
votes
1answer
307 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 ...
7
votes
1answer
45 views

prove the sequence is increasing

Im asked to show the sequence $a_{n+1}=\sqrt{3+2a_{n}}$ where $a_{1}=0, a_{2}=1$ is increasing and bounded and therefore convergent. I don't even know how to start the proof. Im sure it increases ...
7
votes
1answer
246 views

Complicated Multivariate Recurrence Relations For Generating Polynomials

I have the following multivariate recurrence relations all from the same system: First, suppose that $0\le k\le j\le m$, and let $N$ be an independent integer. Then we have for expressions $a(k,~ m,~ ...
7
votes
1answer
309 views

How to prove this sequence of inequalities

The number $c_{g}(n)$ is defined by the recurrence \begin{equation} c_{g}(n) = c_{g}(n-1)+ (n-1)(n-2)c_{g-1}(n-2) , \end{equation} with $c_{0}(n)=1$ for any $n\geq 1$ and $c_{g}(n)=0$ if $n \leq 2g$. ...
7
votes
2answers
923 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 ...
7
votes
1answer
214 views

Diagonal of the double sequence $(n+1)v_{h,n+1}-(2h+1)v_{h,n}-nv_{h,n-1}=0$

Update: it is not possible to reply to this question without additional information. My comment below: "I have to agree with you that one "cannot derive (2) from (1) alone". Now it seems to me that ...
7
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1answer
460 views

Generating function of words in a binary alphabet counting blocks and appearances

Given the binary alphabet {a,b}, I'm trying to find the generating function that distinguishes, for all words of fixed length $n$, the count of blocks of a's and the number of a's. Let $x^p$ count the ...
7
votes
1answer
167 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 ...
7
votes
1answer
263 views

Compute limit of the sequence $x_n$ given by $x_{n+2}=-\frac{1}{2}(x_{n+1}-x_n^2)^2+x_n^4$

Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
7
votes
1answer
78 views

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$

Consider the following iterative process. We start with a 2-element set $S_0=\{0,1\}$. At $n^{\text{th}}$ step $(n\ge1)$ we take all non-empty subsets of $S_{n-1}$, then for each subset compute the ...
7
votes
1answer
50 views

Recurrence $u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$

I was going through some old puzzles and encountered one where there was a non-linear recurrence of form $$u_{n+1}u_{n-1}-u_n^2=Ar^{n-1}$$ with $u_0$ and $u_1$ given. I know I can use a test form ...
7
votes
1answer
2k views

Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $

Express recurrence relation of the integral $$ I_n=\int\frac{dx}{(1+x^2)^n} $$ [My Answer] $$ I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx $$ $$ I_n=I_{n-1}-\int ...
7
votes
1answer
627 views

The Average Running Time Of Euclid Algorithm?

What is the average running time of Euclid Algorithm with respect to all possible input pairs $(m,n)$ such that $\gcd(m,n) = d$? It seems very hard to deduce from the recurrence $T(m,n) = T(n, m ...
7
votes
1answer
703 views

Any serious work on Lychrel numbers/$196$-Algorithm?

I've been googling around a little lately and have stumbled across the so called Lychrel problem. For a natural number $x$, let $Rx$ denote the number obtained by reversing the base-$10$ digits of $x$ ...
7
votes
0answers
244 views

Is there a way to write this recurrence relation in faster-to-program manner?

I have the following recurrence relation for some coefficients $$b_{n+2} = \frac{1}{(n+3)(n+2)P_0} \sum_{k=1}^n (n-k+2) (n-k+1) b_k b_{n-k+2}, \quad n>1$$ with $b_1$ to $b_3$ and $P_0$ being the ...
7
votes
0answers
41 views

Bounds (and range) of a nonlinear difference equation

I'm interested in the following set of nonlinear difference equations: $$x_{n+1} = \frac{c + x_n}{x_{n-1}},\; x_1 = x_0 = 1 \qquad \textrm{for } c > 0$$ For $c=1$ the sequence is periodic with ...
7
votes
0answers
62 views

Asymptotic value of card drawing game

A deck consisting of $r_0$ red cards and $b_0$ black cards is randomly shuffled. The host turns up the cards one at a time; if it is red, you get $\$1$; otherwise you pay the host $\$1$ (and you're ...
7
votes
0answers
125 views

An Impossible Sequence of Prime Powers

Let $x_1,x_2,\ldots$ be a sequence of positive integers that satisfies the recurrence relation $$x_{n+1}=2x_n(x_n-1)+1$$ for all positive integers $n$. It seems impossible that every term in this ...
7
votes
0answers
119 views

Minimizing beginning terms of a Fibonacci-like sequence

For $ a, b \in \mathbb{Z}^+ $ such that $ a = b $ or $ 2a < b $, let $ f_{a, b} : \mathbb{Z}^+ \to \mathbb{Z}^+ $ be a sequence that satisfies the following three criteria: $$ f(1) = a, \\ f(2) = ...
7
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0answers
71 views

Finding a recurrence relation for a sequence not containing $0$ in the digits. Also showing $\sum\limits_{k=1}^n \dfrac{1}{a_k} < 90$

Let $1,2,3,4,5,6,7,8,9,11,12,\ldots$ be the sequence of all positive integers which do not contain the digit zero. Write $\{a_n\}$ for this sequence. By comparing with a geometric series, show that ...
7
votes
0answers
178 views

General overview about Recursion (free online texts)

I'm looking for free online texts about recursion. What I'm looking for formal definitions* of "all" (most of) the different types of recursion and from different points of views like Category ...
7
votes
1answer
160 views

Another recurrence… $T(n)=\sqrt{2n} \cdot T(\sqrt{2n}) + \sqrt{n}$

I'm trying to solve the following recurrence : $$T(n)=\sqrt{2n}\cdot T(\sqrt{2n}) + \sqrt{n}$$ I've tried substituting $n$ for some other variables to transform the above to something easier without ...
7
votes
0answers
107 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
6
votes
8answers
1k views

Need help deriving recurrence relation for even-valued Fibonacci numbers.

That would be every third Fibonacci number, e.g. $0, 2, 8, 34, 144, 610, 2584, 10946,...$ Empirically one can check that: $a(n) = 4a(n-1) + a(n-2)$ where $a(-1) = 2, a(0) = 0$. If $f(n)$ is ...
6
votes
4answers
474 views

How to solve this recurrence $K(n)=2K(n-1)-K(n-2)+C$?

The recurrence is $K(n)=2K(n-1)-K(n-2)+C$ where $C$ is a constant. What I have tried is substituting $2K(n-1)$ as we do in fibonnacical recurrences. It didn't gave me a fruitful expression! Can ...
6
votes
6answers
2k views

How to find $\lim a_n$ if $ a_{n+1}=a_n+\frac{a_n-1}{n^2-1} $ for every $n\ge2$

$a_n$ is a sequence where $a_1=0$ and $a_2=100$, and for $n \geq 2$: $$ a_{n+1}=a_n+\frac{a_n-1}{(n)^2-1} $$ I have a basic understanding of sequences. I wasn't sure how to deal with this recurrence ...
6
votes
2answers
2k views

Recursive square root problem [duplicate]

Give a precise meaning to evaluate the following: $$\large{\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dotsb}}}}}$$ Since I think it has a recursive structure (does it?), I reduce the equation to $$ ...
6
votes
4answers
150 views

closed form for $d(4)=2$, $d(n+1)=d(n)+n-1$?

I am helping a friend in his last year of high school with his math class. They are studying recurrences and proof by inference. One of the exercises was simply "How many diagonals does a regular ...
6
votes
5answers
258 views

All the ternary n-words with an even sum of digits and a zero.

I'm trying to find a recursive formula for all the ternary (using ${0,1,2}$) sequences of length $n$ which contain at least one zero, and have an even sum of digits. My attempt so far is added below. ...
6
votes
5answers
130 views

Recurrence of the form $2f(n) = f(n+1)+f(n-1)+3$

Can anyone suggest a shortcut to solving recurrences of the form, for example: $2f(n) = f(n+1)+f(n-1)+3$, with $f(1)=f(-1)=0$ Sure, the homogenous solution can be solved by looking at the ...
6
votes
3answers
151 views

Prove that $A_{100} \gt 14$ where $A_{n}=A_{n-1}+\frac{1}{A_{n-1}}$ and $A_1=1$

I tried attempting the question, and the best upper bound I could obtain was $1+\ln{98}$. I tried using $A_{n}\le n$ to form a harmonic series, but that wasn't strong enough. Any help would be ...
6
votes
2answers
78 views

How to evaluate $\lim_{ n\to \infty }\frac{a_n}{2^{n-1}}$, if $a_0=0$ and $a_{n+1}=a_n+\sqrt{a_n^2+1}$?

Let $a_1,a_2,..,a_n$ be sequence of real numbers such that $a_{n+1}=a_{n}+\sqrt{1+a_n^2}$ and $a_0=0$. How to evaluate $\lim_{ n\to \infty }\frac{a_n}{2^{n-1}}$ ?
6
votes
2answers
231 views

A recurrence relation problem: $\frac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \frac 1 n$

I need to solve this recurrence problem to find $a_n$ $\dfrac {a_{n-1}.a_{n+1}} {a_n^2} = 1 + \dfrac 1 n$ It is what I tried so far: $$\log (\dfrac {a_{n-1}.a_{n+1}} {a_n^2}) = \log(1 + \dfrac 1 ...
6
votes
1answer
209 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 ...
6
votes
5answers
159 views

$x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$ Proof

Prove $x_{n+1} = \sqrt{3x_n}$ converges for $x_1 = 1,x_1 = 27$. (Separate problems for $x_1 = 1$ and $x_1 = 27$.) EDIT: Took out bad algebra.
6
votes
5answers
170 views

Convergence of a recurrence

Given the recursive definition (starting with a positive integer) $$ a_n = \frac{a_{n-1}}{2}+4 $$ I am trying to find an explicit form and show that it approaches 8. So I started by writing it out, ...
6
votes
3answers
93 views

Please solve this recurrence relation question for $8a_na_{n+1}-16a_{n+1}+2a_n+5=0$

Suppose $a_1=1$ and $$8a_na_{n+1}-16a_{n+1}+2a_n+5=0,\forall n\geq1,$$Please help to sort out the general form of $a_n$. Here are the first a few values of the series. Not sure if they are useful as ...
6
votes
4answers
285 views

Alternating Recurrence relation $a_n = b_{n-1} + 5$ and $b_n = na_{n-1}$

I am racking my brain on solving the relation where: $$a_n = b_{n-1} + 5$$ $$b_n = na_{n-1}$$ where $a_0$ = $b_0$ = 1 I am trying to find the closed form for $a_n$. I have tried to shifting $b_n = ...
6
votes
4answers
234 views

Recursive Sequence Tree Problem (Original Research in the Field of Comp. Sci)

This question appears also in http://cstheory.stackexchange.com/questions/17953/recursive-sequence-tree-problem-original-research-in-the-field-of-comp-sci. I was told that cross-posting in this ...
6
votes
4answers
384 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, ...
6
votes
3answers
1k 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 ...
6
votes
4answers
2k views

Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
6
votes
1answer
2k views

Concrete Mathematics - Towers of Hanoi Recurrence Relation

I've decided to dive in Concrete Mathematics despite only doing a couple of years of undergraduate maths many years ago. I'm looking to work through all the material whilst plugging gaps in my ...
6
votes
1answer
5k 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
1answer
912 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
1answer
601 views

convoluted recurrence: $f(2n)=f(n)+f(n+1)+n, f(2n+1)=f(n)+f(n-1)+1$

For the recurrence relation: $f(0)=1$ $f(1)=1$ $f(2)=2$ $f(2n)=f(n)+f(n+1)+n\ \ \ (\forall n>1)$ $f(2n+1)=f(n)+f(n-1)+1\ \ \ (\forall n\ge 1)$ (the first numbers of the sequence are: 1, 1, 2, ...
6
votes
2answers
589 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
1answer
5k views

What are linear homogeneous and non-homoegenous recurrence relations?

According to my book, linear homogeneous of order k is expressed this way: $$A_0a_n+A_1a_{n-1}+A_2a_{n-2}+\cdots+A_ka_{n-k}=0$$ While a linear non-homoegeous of order k is this way: ...
6
votes
4answers
1k 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 ...