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

learn more… | top users | synonyms (2)

6
votes
1answer
90 views

How can I find $f(3)$ if $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$?

If $f(2x)=f^2(x)-2f(x)-\frac{1}{2}$ and $f(1) = 2$ then find $f(3)$. Can you give me any hint that I can start with?
6
votes
1answer
83 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
6
votes
1answer
110 views

A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$

I am not sure whether this is right. Can anyone verify, whether this proof is valid? Thanks! Define a sequence $\{a_n\}_{n\ge0}$ as follows: ...
6
votes
2answers
378 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
183 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
2k views

Solving recurrence $T(n) = T(\lceil n/2 \rceil) + T(\lfloor n/2 \rfloor) + \Theta(n)$

I'm learning algorithms by myself and am using the excellent Introduction to algorithms to do that. It has been quite a long time since I last studied math, so maybe the solution to my problem is ...
6
votes
1answer
56 views

The number of series over $\{0,1,2\}$ without repeating numbers

What is the number of series over $\{0,1,2\}$ with length $n$ without repeating the same number one after the other ($22$ is not allowed but $101$ is), that does not begin and end with the number $2$. ...
6
votes
1answer
305 views

Recursion relation of fourth order Runge-Kutta method applied on system

I'm trying to apply the Gauss-Legendre method of fourth order (as Runge-Kutta method) on the following system of equations $$\left\{ \begin{matrix} \dot{a} =& -b \\ ...
6
votes
1answer
155 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
50 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 ...
6
votes
0answers
228 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 ...
6
votes
0answers
50 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 ...
6
votes
0answers
79 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
votes
7answers
149 views

Convergence of the recursive sequence $a_{n+1}=\sqrt{2-a_n}$

I'm attempting to prove the convergence (or divergence, though I strongly suspect it converges) of a sequence defined as $a_{n+1}=\sqrt{2-a_n}$ with $a_1=\sqrt{2}$. I cannot use the monotonic ...
5
votes
5answers
334 views

Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$

$T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $ Show that $T(n) = 2^n$. I know how to prove this by induction, but I would like to know how to show this using first principles. Edit: The way I want ...
5
votes
4answers
312 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
542 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
3answers
1k views

Determining if a recursively defined sequence converges and finding its limit

Define $\lbrace x_n \rbrace$ by $$x_1=1, x_2=3; x_{n+2}=\frac{x_{n+1}+2x_{n}}{3} \text{if} \, n\ge 1$$ The instructions are to determine if this sequence exists and to find the limit if it exists. I ...
5
votes
2answers
220 views

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

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
5
votes
3answers
408 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
2answers
418 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
3answers
966 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
13k views

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

How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part.
5
votes
5answers
1k 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
4answers
469 views

How to find the general formula for this recursive problem?

I want to ask about recursive problem. Given: $$a_0= 11, a_1= -13,$$ and $$a_n= -a_{n-1} +2a_{n-2}.$$ What is the general formula for $$a_n$$ ? I've already tried to find the first terms of this ...
5
votes
2answers
187 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
5answers
344 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
3answers
140 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
84 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
3answers
2k views

Recurrence trouble: $T(n)=2T(n/2)+T(n/3)+\theta(n^2)$

I have to solve the following recurrence :$\displaystyle T(n)=2T(n/2)+T(n/3)+\theta(n^2)$ I have done the whole tree analyses and now I have to prove that $\displaystyle T(n) \leq ...
5
votes
3answers
286 views

Mathematical Induction proof for a cubic equation.

If $ x^3 = x +1$, prove by induction that $ x^{3n} = a_{n}x + b_n + \frac {c_n}{x}$, where $a_1=1, b_1=1, c_1=0$ and $a_n = a_{n-1} + b_{n-1}, b_n = a_{n-1} + b_{n-1} + c_{n-1}, c_n = a_{n-1} + ...
5
votes
1answer
178 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
4answers
661 views

How to solve this recurrence relation?

There's a frog who could climb either 1 stair or 3 stairs in one shot. In how many ways he could reach at 100th stair? I came up with the solution $g(n) = g(n-3) + g(n-1)$, where $g(0)=g(1)=g(2)=1$ ...
5
votes
2answers
299 views

Sum of the series formula

I need to figure out the sum of the series as quickly as possible in a program given n and k: $$f(n,k)= ...
5
votes
3answers
86 views

What are the possible limits of the iteration $x_{n+1}=\sqrt{x_n+3}$, $x_0=0$?

Let $f(x)=\sqrt{x+3}$ for $x\ge -3$. Consider the iteration $$x_{n+1}=f(x_n),x_0=0;n\ge 0$$ The possible limits of the iteration are -1 3 0 $\sqrt{3+\sqrt{3+\sqrt{3+\cdots}}}$ I think only ...
5
votes
3answers
181 views

How does $\tbinom{4n}{2n}$ relate to $\tbinom{2n}{n}$?

I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation: $$(2n)!=\binom{2n}{n}(n!)^2$$ which made me wonder: is there also a ...
5
votes
2answers
338 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
3answers
619 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
1answer
162 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
1answer
381 views

Expanding the generating function of the Fibonacci numbers to find a cute formula

$F_0=1$, $F_1=1$, $F_n=F_{n-1}+F_{n-2}$. The generating function is $-\frac{1}{x^2+x-1}$. I have to expand it to prove that $F_n=\sum_k\binom{k}{n-k}$. Could you help me please?
5
votes
2answers
139 views

Is it possible to compute factorials by converting to matrix multiplications?

An $n$-th term of the Fibonacci sequence can be computed by a nice trick by converting the recurrence relation in a matrix form. Then we compute $M^n$ in $O(\log n)$ steps using exponentiation by ...
5
votes
2answers
134 views

Proving integrality of a sequence of numbers

How would one go about proving whether or not the terms of a sequence such as the following $$1, 1, 1, 1, 1, 6, 21, 181, 5221, 1090981, 986241401, 51490676208426,\ldots$$ defined by ...
5
votes
2answers
131 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
3answers
183 views

Need help about solving a recurrence relation

I have a recurrence relation which is like the following: $$ T(n) = 2T(n/2) + \log_2 n. $$ I am using recursion tree method to solve this. And at the end, i came up with the following equation: $$ ...
5
votes
1answer
211 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
1answer
732 views

On the generating function of the Fibonacci numbers

Let's define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be ...
5
votes
3answers
146 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
144 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
2k 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
2answers
182 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, ...