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

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5
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3answers
143 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into ...
5
votes
4answers
137 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
173 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
190 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 $a_1=\frac{1}{2}, a_2=\frac{1}{4}$
5
votes
2answers
416 views

Whats better: 1 million dollars in a month or a penny(USD) doubled (and added) every day for 30 days?

THis is a question that I remember when I was in the 5th grade that tested our logical reasoning skills. And it is a simple choice knowing that the pennies doubling every day is an exponential ...
5
votes
2answers
167 views

Recurrence relation: composition of a polynomial function

Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is $$f_1(x)=f(x)$$ $$f_2(x)=f \circ f(x)$$ $$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$ Is there a way to find a formula for ...
5
votes
1answer
236 views

When is a recurrence relation linear

In http://www.cs.fsu.edu/~lacher/courses/COT5405/spring07/notes2.html, it says that $T(n) = aT(n/b) + f(n) $ is nonlinear recurrence. But I think it is linear because $T(n)$ is linear in $T(n/b)$. ...
5
votes
2answers
1k views

Second order homogeneous linear difference equation with variable coefficients

I was wondering if you would point me to a book where the theory of second order homogeneous linear difference equation with variable coefficients is discussed. I am having difficulties in getting ...
5
votes
3answers
745 views

how to solve $T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}$

How can I solve the recurrence $$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$ Please don't just write the name of the method, as I just started learning this stuff and things are a ...
5
votes
2answers
215 views

Method of solving no-homogeneous recurrence equation

I need to obtain a closed form of $M(t)$, satisfying the following recurrence equation: $$M(t+1)=a+bM(t)+\frac{c}{t+1}\sum_{t'=0}^tM(t')+df(t)$$ Where $f(t)$ is a known function and $a$, $b$, $c$ ...
5
votes
1answer
419 views

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is an exercise from Chapter 1 of "Concrete Mathematics". It concerns the Towers of Hanoi. Are there any starting and ending configurations of $n$ disks on three pegs that are more than $2^n - ...
5
votes
3answers
67 views

Divergence of $a_{n+1}=\sqrt{2a_n+3}$?

I am wondering what I am missing from my proof. I would like to show that the limit of the sequence $$a_{n+1}=\sqrt{2a_n+3},\,\, a_1=4,$$ goes to $\infty$, as $n \rightarrow \infty$. Is there ...
5
votes
2answers
178 views

Let $x_{n+1} = x_n + 1/(x_1 + x_2 +\ldots + x_n)$ with $x_1 = 1$. Show that $x_n\sim\sqrt{2\log(n)}$.

As the title states we have a sequence defined by $$x_{n+1} = x_n + \dfrac{1}{x_1 + x_2 + \cdots + x_n}$$ with $x_1 = 1$. The first few terms are: $1, 2, \frac{7}{3}, \frac{121}{48} \cdots$ Any ...
5
votes
3answers
218 views

Solve the recursion, $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$

Bring the following recursion relation to an explicit expression: $$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$$ $a_{0} = 0$, $a_1 = 1$, $a_2 = 2$ All the examples I have seen were with maximum 2 steps back ...
5
votes
2answers
64 views

Let $A$ be a matrix sized $p\times p$, where $2\le p$. Using recurrence relations, describe $A^k$.

Let $A$ be a matrix sized $p\times p$, Where $2\le p$. The matrix values in the main diagonal are $0$ and the rest are $1$'s. Example for $A$ where $p=5$: $$\begin{bmatrix} 0 & 1 & 1 & 1 ...
5
votes
2answers
583 views

Solving recurrences of the form $T(n)=aT(n/a)+Θ(nlgn)$

On pages 95 and 96 of the third edition of the CLRS book, we find the following, which applies here since $a=b$ is all it takes to block the application of the Master Theorem: "Although $n\lg n$ is ...
5
votes
1answer
273 views

two-dimensional recurrence

Can someone using only these conditions $$a_{m,k}=a_{m-1,k}+a_{m-1,k-1},m>k$$ $$a_{m,k}=1,m=k$$ $$a_{m,k}=0,m<k$$ prove that $$a_{m,k}=\frac{m!}{k!(m-k)!}$$ here is way to construct Pascal ...
5
votes
1answer
451 views

Show that the Catalan numbers are given by this recurrence relation

Hey guys! I'm doing an assignment, and I'm just not sure (at all) how to start this problem. Can somebody nudge/shove me in the right directions? Show that the Catalan numbers are given by the ...
5
votes
1answer
159 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,~ ...
5
votes
1answer
141 views

What is the expected number of questions answered to complete a sequence in which wrong answers send you to the start?

Given a sequence of n questions that each contain x answer choices, what is the expected number of questions answered before answering all questions correctly if answering a question incorrectly sends ...
5
votes
1answer
130 views

Period doubling bifurcation in a quadratic map

I am attempting the find $\mu$ for which the map $$x_{n+1} = \mu + x_n^2$$ undergoes a period doubling bifurcation. I understand that finding the fixed points of the map is the first step towards ...
5
votes
1answer
278 views

Recurrence for perfect matchings revisited.

I like to study combinatorics a bit as a hobby, and recently a question I found interesting was posed asking to derive a recurrence for the generating function $G_n(x)$, the ordinary generating ...
5
votes
2answers
84 views

Proving a solution to a double recurrence is exhaustive

The equation $$ b^2 = \frac{a(a+1)}{2} + 1 $$ where $a$ and $b$ are integers, has the following smallish-integer solutions: ...
5
votes
1answer
88 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
5
votes
3answers
293 views

How to solve this recurrence $T(n)=2T(n/2)+n/\log n$

How can I solve the recurrence relation $$T(n)=2T\left(\frac n2\right)+\frac{n}{\log n}$$? I am stuck up after few steps.. I arrive till $$T(n) = 2^k T(1) + \sum_{i=0}^{\log(n-1)} ...
5
votes
1answer
472 views

Deriving a recurrence relation

The number of sequences of length $n$ consisting of positive integers such that the opening and ending elements are $1$ or $2$ and the absolute difference between any $2$ consecutive elements is $0$ ...
5
votes
1answer
111 views

Geometric interpretation of the fundamental theorem for coalgebras?

Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the ...
5
votes
1answer
166 views

general technique to convert recurrence relation to integral

I know the following recurrence relation $$a_n=\frac{a+na_{n-1}}{a-n}$$ with $a_0=1$ can be represented alternatively as an integral $$a_n=a\int_0^1{x^{a-n-1}(2-x)^ndx}$$ Verifying this is easy, ...
5
votes
1answer
251 views

A nicer recurrence for the Eulerian polynomials.

I was perusing the subject of Eulerian polynomials. I'm assuming the definition that the Eulerian polynomial is defined by $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of ...
5
votes
2answers
131 views

How can I prove that this recursive sequence converges?

Let $a_0=1$, $a_1=1$ and $a_{n+2}=\frac{1}{a_{n+1}}+\frac{1}{a_n}$. How can I prove that this sequence is convergent? I know that if it's convergent, it converges to $\sqrt{2}$ and I can calculate the ...
5
votes
1answer
190 views

Solving a recurrence with diagonalization?

Considering the recurrence $F_n=F_{n-1}+3F_{n-2}-3F_{n-3}$ where $F_0=0$, $F_1=1$ and $F_2=2$. Use diagonalization to find a closed form expression for $F_n$. So I first continued the recurrence to ...
5
votes
1answer
60 views

Recursive identity for elliptic lattice constants $\sum_{\lambda\in\Lambda\setminus0} \lambda^{-2k}$

I am stuck on Exercise 3 in these notes. To keep this question self-contained: we have $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
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4answers
116 views

Homework | Find the general solution to the recurrence relation

A question I have been stuck on for quite a while is the following Find the general solution to the recurrence relation $$a_n = ba_{n-1} - b^2a_{n-2}$$ Where $b \gt 0$ is a constant. I don't ...
5
votes
1answer
108 views

Recurrence for random walk

I have the following recurrence which I get when trying to solve a random walk problem given a positive integer $x$. $p_i = \dfrac{p_{i-1}}{2} + \dfrac{p_{i+2}}{2}$ if $0< i < x$ $p_i = 1$ if ...
5
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1answer
404 views

Purely combinatorial proof and simplification of identity involving factorials and summations

While trying to decompose factorials into summations, I came up with the following identity $$(n+2)! = 2^{n+1} + \sum\limits_{k=0}^{n-1}\sum\limits_{i=0}^{n-1-k}\sum\limits_{S \subseteq ...
5
votes
3answers
327 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...
5
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1answer
82 views

Recurrence-differential equation

In his book on differential equations, Arnold writes that $x'(t)=x(x(t))$ is not a differential equation. My question is: how can one solve it?
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1answer
939 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 ...
5
votes
1answer
79 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 ...
5
votes
1answer
85 views

Does the A001921 linear recurrent integer sequence always yield composite numbers?

Let $(a_n)$ be the A001921 sequence $$ a_0 := 0,\ a_1 := 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6. $$ Is it true that $a_n$ is always a composite integer for any $n\geq 2$ ? UPDATE : I now make a ...
5
votes
0answers
143 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...
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0answers
44 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 ...
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0answers
70 views

Right way to solve this reccurence relation?

I am not that good in mathematics , I have solved many simple relations. But I am not be able to solve the following. This is not a homework. I found it during analysis of a program. ...
5
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0answers
166 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 ...
5
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0answers
159 views

number of potential couples

A potential couple is a pair of a man and a woman that like each other (assume that 'like' is a symmetric relation). Given a group of $M$ men and $W$ women, I want to know how many different ...
4
votes
5answers
419 views

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$,$F(1)=b$ and $a,b>0$?

What is closed-form expression for $F(n)$ when $F(n)=F(n-1)+F(n-2)$ and $F(0)=a$, $F(1)=b$ and $a,b>0$ ? It seems to be simple generalization of Fibonacci sequence but I can't find closed form for ...
4
votes
4answers
329 views

What's known about recurrences involving $(a_n)^2$?

I've run across the recurrence $a_{n+1} = (a_n)^2 + 1$ in the past. Unfortunately, the referrence escapes me. However, my impression was that recurrences involving the product of previous terms ...
4
votes
4answers
234 views

Recurrence relation $C_n = n + 1 + \dfrac{2}{n}\sum\limits_{k=0}^{n-1}C_k$.

A Discrete Mathematics book from which I'm self-studying ("Discrete Mathematics and Its Applications", by Kenneth Rosen) asks me to do the following: Given the following recurrence relation: $$C_n = ...
4
votes
3answers
1k views

Notation for factorial-type pattern with a skip/step of two instead of one?

I came across a peculiar pattern when solving a recurrence relation today: Some sequence $a_n$ looks as such: $a_0 = 1$ $a_2 = \frac{1}{2 \cdot 1}$ $a_4 = \frac{1}{4 \cdot 2 \cdot 1}$ $a_6 = ...
4
votes
4answers
266 views

Finding the billionth number in the series: $2, 3, 4, 6, 9, 13, 19, 28, 42, \ldots $?

Series is defined as $$a_{n+1} = \lfloor\frac{3\cdot a_n}{2}\rfloor,\qquad a_0 = 2$$ It can be viewed as the number of animals starting from a single pair if any pair of animals can produce a single ...