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

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Inductive proof of the closed formula for the Fibonacci sequence [duplicate]

Fibonacci numbers $f(n)$ are defined recursively: $f(n) = f(n-1) +f(n-2)$ for $n > 2$ and $f(1) = 1$, $f(2) = 1$. They also admit a simple closed form: $$\sqrt 5 f( n ) = \left(\dfrac{1+ \sqrt 5}{...
7
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1answer
252 views

Finding $a_n$ for very large $n$ where $a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $

I have a recurrence relation, $$ a_n = a_{n-1} + a_{n-2} + a_{n-3} + 2^{n-3} $$ for $n>3$ and $a_1 = 0, a_2 = 0, a_3 = 1$ I have to find the value of $a_n$ for very large values of n. I tried ...
7
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3answers
98 views

How to prove that the recurrence $a_{n}=a_{n-1}+n^2a_{n-2}$ gives $(n+1)!$ without induction

Define the sequence $\{a_n\}$ by $a_{1}=2,a_{2}=6$, and for $n>2$, $$a_{n}=a_{n-1}+n^2a_{n-2}$$ show that $$a_{n}=(n+1)!$$ I know if we use induction,it is easy to prove it. $$n!+n^2(n-1)!=n![...
7
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2answers
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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, ...
7
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1answer
377 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: $$f(n)=f\left(\left\lfloor\frac{2n-1}{3}\right\rfloor\...
7
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1answer
424 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 ...
7
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1answer
105 views

$2005$th derivative of $f$ at $0$

So I tried using Leibnitz formula to solve by recurrence, but I can just get to one point and then it's a mess again. Problem is Let $f(x)=\frac{1}{1+2x+3x^2+\ldots+2005x^{2004}}$. Find $f^{[2005]}...
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1answer
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Adapted Towers of Hanoi from Concrete Mathematics - number of arrangements

I have a doubt concerning an exercise from Chapter 1 of "Concrete Mathematics". Actually, my doubt is in one exercise (exercise 3), but, since it depends on the previous exercise (2), I'm including it ...
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3answers
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Solving recurrence relation of form $T(n/2 + c)$

It is obvious that the Master Theorem cannot be applied to the recurrences of the following form: $T(n) = 4T(n/2 + 2) + n$ Since I am only interested in the $\theta$ bound of the recurrence and not ...
7
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1answer
134 views

Put a mouse to the last cell

We have (n=12) cells $C_1, C_2 ,\dots, C_{12}$ which are initially empty. At each step, we can do one of two operations: $\mathbf{P}$: Put only in the first cell $C_1$ 2 mice. $\mathbf{M}$: Move ...
7
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2answers
322 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
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1answer
309 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 ...
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1answer
47 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
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1answer
258 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,~ ...
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1answer
310 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
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2answers
939 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 ...
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1answer
215 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 ...
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1answer
461 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 ...
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1answer
169 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 $$\sqrt{S(x)+\sqrt{S(x+1)+\sqrt{....
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1answer
267 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 \dfrac{3}{4},\...
7
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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
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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 x\cdot\frac{x}{...
7
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1answer
628 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 \...
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1answer
705 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 $x$ ...
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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
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0answers
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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 ...
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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 ...
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0answers
128 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 ...
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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) = ...
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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 $\...
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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 ...
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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 ...
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0answers
108 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 ...
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8answers
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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 $\...
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4answers
478 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 ...
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6answers
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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 ...
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2answers
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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 $$ p=\...
6
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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 $n$-...
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5answers
263 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. ...
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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
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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
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2answers
79 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
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2answers
239 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 n)$...
6
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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
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5answers
160 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
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5answers
171 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
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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
291 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
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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
391 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, $...