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

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Master Theorem Question

I need to solve the following: $T(n)=T(n-1)+8$ I've tried doing $a=1$, $b=-1$, and $d=8$ but $\log_{-1}1$ doesn't make sense. Any suggestions?
5
votes
1answer
280 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 ...
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1answer
27 views

Recurrence relation for $n$ digit numbers not containing '$20$'

How many n digits base $3$ numbers do exist such that they never contain pattern '$20$'? (first find a recurrence relation)
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2answers
58 views

recursion-consecutive numbers

what is the number of subsets of the set {k∈N|1≤k≤n} with no two consecutive numbers? The answer says: $$a_n=a_{n-1}+a_{n-2}$$ with the starting conditions: $$a_0=1, a_1=2$$1. why does $a_1=2$? $$$$ ...
2
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0answers
85 views

How to solve a ceiling expression or recurrence equation?

I am trying to express the following in term of n, $$\underbrace{\hbox{$ \left \lceil \frac{3}{2}.\left \lceil { \frac{3}{2}.\left \lceil { \ldots \left \lceil \frac{3}{2}.\left \lceil { ...
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votes
1answer
24 views

Characteristic equation of a difference equation indicates the function behavior

For the characteristic equation $a_2 \lambda^2 + a_1 \lambda + a_0 = 0$ of the difference equation $a_2 x_{n+2} + a_1 x_{n+1} + a_0 x_n = 0$, I remember there is a way to indicate if the function of ...
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4answers
37 views

Identity for this recursive relation with multiple terms

I have a recursive relation algorithm which is defined as follows: $$F_n = 3(F_{n-1} - F_{n-2}) + F_{n-3}$$ $$F_0 = 0$$ $$F_1 = 1$$ $$F_2 = 4$$ From calculating the first few values, I know this is ...
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1answer
19 views

How do we solve a tight big-O bound for the recurrence $T(n) = T(n^{2/3}) + 1$?

The big-O bound seems largely governed by how many times we can take the input $n$ by the $\frac{2}{3}$ power until it reaches some constant like 1. How do I start formalizing this problem in math ...
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2answers
47 views

How do we solve the exact recurrence for $T(1) = 1, T(n) = 3T(n - 1) + 2n + 2$ for $n > 1$?

This looks like an exponential recurrence due to the 3 behind $T$, but I'm not sure how to formally solve for $T(n)$ without $T$ on the righthand side.
2
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1answer
29 views

Four or More Heads in a Coin Toss

How would one write a recursion for the number of ways to get 4 or more heads in a row in 10 tosses of a coin? Would it suffice to use the recursion H(n)=H(n-1)+H(n-2)+H(n-3)+H(n-4) for the number of ...
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0answers
27 views

How many bit strings of length n has two consecutive 0's? [duplicate]

After little bit of Analysis of bit strings of length 2,3,4,5. I come across this recurrence for the above problem. $$a_n = 2 a_{n-1} + (n-2)$$ Is it right ? If yes then please share more simple ...
3
votes
3answers
73 views

Why does this recurrence relation generate a sinusoidal curve?

I came across the following coupled recurrence relation while watching this video called Media for Thinking the Unthinkable: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
11
votes
1answer
178 views

Family of GCDs all equal to $2$

Is it true that $$\gcd\left(5^{2^n} + 1, 13^{2^n} + 1\right) = 2$$ for all $n \in \mathbf{Z}_{\geq 0}$? I'm continually stumped with this and verifying it numerically is quite expensive very ...
0
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1answer
112 views

Solving a recurrence involving floor and square root (Concrete Mathematics 3.28)

I'm working through Concrete Mathematics and having trouble understanding an answer to a problem (as well as what I could've done to come up with the answer). Problem 3.28 asks: Solve the recurrence ...
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2answers
84 views

Combinatorics - Check my answer, sitting order, round table.

$n$ people are sitting at a round table with $n$ seats at a restaurant. The restaurant has only 2 dishes, steak and salad. How many ways are there for the diners to choose a dish, such that no 2 ...
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1answer
100 views

Solve Recurrence Equation with Induction

Question: Given the recurrence equation for the recursive Fibonacci sequence program: $T(n) = T(n-1) + T(n-2) + b$ $T(0) = T(1) = a$ Using induction, show that $T(n) \leq f(n)$, where $f(n) = c2^n, ...
3
votes
3answers
91 views

Recurrence $f(a,b)=f(a,b-1)+2f(a-1,b-1)$

Consider the recurrence relation $$f(a,b)=f(a,b-1)+2f(a-1,b-1)$$ for integers $a,b\geq 2$, where $f(a,b)=1$ if $a=1$ or $b=1$. Is it possible to find a closed form for $f(a,b)$?
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1answer
54 views

number of ways to fill a 2D grid

We have a 2D grid with n rows and m columns, we can fill it with numbers between 1 and k (both inclusive). Only condition is that for each r such that 1<=r<=k ,no two rows must have exactly the ...
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1answer
48 views

Can every recurrence relation be solved?

Motivation A possible way to solve an ODE is to express the solution as: $y= \sum_{n=0}^\infty a_nx^n$. We substitute in the ODE and then calculate the coefficients $a_n$. For example, $y''+y=0$ ...
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1answer
43 views

Master's theorem applicability.

I have to find out if the following recurrence can be solved with the master theorem: $$T(n) = 3T\left(\frac{n}{2}\right) + n^{\log\log n}$$ I have figured that, here, I have the third case because ...
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1answer
37 views

Asymptotic upper bound for recurrence relation

Obtain asymptotic upper bound for recurrence relation $$T(n)=4T(\frac n2)+n^2 \log n$$ Am not getting the answer. Plz help me
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1answer
85 views

if $(n+1)c_{n}=nc_{n-1}+2nd_{n-1},d_{n}=2c_{n-1}+d_{n-1}$How prove $c_{n},d_{n}$ are integers

let sequences $c_{n},d_{n}$ such $$c_{0}=0,d_{0}=1$$ and for $n\ge 1$,have $$c_{n}=\dfrac{n}{n+1}c_{n-1}+\dfrac{2n}{n+1}d_{n-1}$$ $$d_{n}=2c_{n-1}+d_{n-1}$$ show that $c_{n},d_{n}$ are integers. my ...
0
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1answer
47 views

Plot of recurring system in MATLAB, Lozi map

I need to write this recurring system in MATLAB $$ x_{n+1}=1-a|x_n|+y_n$$ $$ y_{n+1}=bx_n $$ and take its plot for every $x_i,y_i$,with let's say a=1.4 and b=0.7. $$$$This is the Lozi map. And this ...
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0answers
30 views

Recurrence of a function

Consider the following recurrence: $$ T(n) = 2T(n/2)+n\lg n$$ Let’s use a base-case $T (2) = 2$ and let’s assume $n$ is a power of $2$. (a) “guess and prove by induction” method, considering the ...
2
votes
3answers
44 views

Help with recurrence relation

It 's been a long time since I touched this kind of math , so please help me to solve the relation step by steps : $V_k = (1+i)*V_{k-1}+P$ I know the answer is $V_k = (P/i)*((1+i)^k-1) $ Thanks ...
4
votes
0answers
147 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 ...
3
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1answer
25 views

Analyzing $n_{i+1} = n_i - n_i^{3/4}$

I have a non-linear recurrence given by $$n_0 = N \\ n_{i+1} = n_i - n_i^{3/4}$$ Are there any techniques to solve this for an exact closed form? Or in lieu of that, an asymptotic estimation? I'm ...
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0answers
31 views

Algorithms: Recurrence

Here's a problem that I am struggling with... If two algorithms A and B both solve the same problem. On an input of size $n$ Algorithm $A$ breaks it into $5$ pieces of size $n/2$, recursively solves ...
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1answer
111 views

Stable algorithms from a backwards recurrence?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of an integral, but the algorithm is unstable when $|\alpha ...
2
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2answers
102 views

Internet problem solving contest question

I am trying to solve a problem from the IPSC http://ipsc.ksp.sk/2001/real/problems/f.html It basically asks to compute the following recursion. ...
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2answers
103 views

Question about the “master theorem” of recurrences - no “$b$” term

I'm using the master theorem to find the asymptotic run time of recurrences. For example, for a $T(n) = 4 T(n/5) + n^1$ I find that $T(n)$ is $\Theta(n^1)$, or, simply, constant time, via the set of ...
2
votes
2answers
53 views

Question with recurrence - Check my answer and suggest better ideas

Every morning when he gets up, Oria leaves the house for a walk. he walks exactly $n$ steps. He can walk only forward, right, or left, and he will never turn left immediately after he turned right, ...
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4answers
86 views

Proving convergence of a sequence

Let the following recursively defined sequence: $a_{n+1}=\frac{1}{2} a_n +2,$ $a_1=\dfrac{1}{2}$. Prove that $a_n$ converges to 4 by subtracting 4 from both sides. When I do that, I get: ...
0
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2answers
44 views

Recurrence relation - Show that a sum of a sequence is zero

We are given the following sequence: $f(n)=4f(n-1)-5f(n-2)$, $f(0)=f(1)=a$ where $a$ is some value in $\mathbb C$. We are asked to show that $$\sum_{n=0}^{\infty}\frac{f(n)}{3^n}=0$$ First thing I ...
2
votes
3answers
82 views

Solving recurrence relation with generating functions - Nearly got the answer

I'm trying to solve the following recurrence relation (Find closed formula) using generating functions: $f(n)=10f(n-1)-25f(n-2)$, $f(0)=0$, $f(1)=1$ I'm having a small difficulty at the end and can ...
0
votes
1answer
187 views

Summation of logarithmic series

I am solving a recurrence relation and it requires me to sum the following series upto $\log{n}$ terms - $1/\log(n) + 1/\log(n/2) + 1/\log(n/4)$..... The base in each term is $2$. Any help on ...
2
votes
0answers
134 views

Special map $f(x_n)=x_{{n+1} [\mod 4]}$

As mentioned in the headline I am considering the map $f(x_n)=x_{{n+1} [\mod 4]}$. It looks a little bit similar to the dyadic (bernoulli) map but instead of $\mod 1$ we have $\mod 4$ which means that ...
4
votes
1answer
65 views

Find recursive forumula for integrals

I have to find recursve formulas for solving the following two integrals. The assignment tells one to find an Expression that leads from the calculation of $\dfrac{I_{2n}}{I_{2n+1}}$ to the ...
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3answers
84 views

$W_n=\int_0^{\pi/2}\sin^n(x)\,dx$ Find a relation between $W_{n+2}$ and $W_n$

Set $$W_n=\int_0^{\pi/2}\sin^n(x)\,dx.$$ Compute $W_0$ and $W_1$. Find a relation between $W_n$ and $W_{n+2}$ and deduce a formula for $W_n$. What I have so far is: $$W_{2k}=\frac{1}{2^k}\left( ...
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1answer
70 views

Prove any $k\in\mathbb{N}$ can be created using sum of $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$

I need to prove that any integer can created using a sum of elements in $a_{n}=2a_{n-1}-\left\lfloor \frac{a_{n-1}}{2}\right\rfloor ,a_{1}=1$ without repetition. My attempt was just bad... I've ...
0
votes
1answer
51 views

Period of recurrence relation mod p

For a recurrence relation like $$f(n)=((k-2)*f(n-1)+(k-1)*f(n-2) )\bmod p$$ with initial conditions .This recurrence holds for n>=4 $$\begin{align}f(1)&=k \bmod p\\ f(2)&=k*(k-1) \bmod ...
4
votes
1answer
111 views

Recursive sequence with binomial coefficients

I have a sequence $\epsilon_i$ defined recursively for $i\ge 1$ as follows \begin{eqnarray*} \epsilon_1 &=& \frac{1}{p}\\ \epsilon_n &=& \frac{1}{1-(1-p)^n}\left( 1 + \sum_{j=1}^{n-1} ...
2
votes
5answers
83 views

Finding a recurrence relation in combinatorics

First, my English is not that good so please don't laugh. I need to find a recurrence relation for : The number of words with the length of n that could be composed from the letters A,B,C,D, so the ...
2
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1answer
43 views

Recursive Definition for Logarithm

Is it possible to express $\log(n)$ in terms of itself, using only elementary functions, e.g., similar to (the incorrect made-up equation): \begin{equation*} \log(n) = e^n * \log(n - 1) + \sin(n) * ...
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2answers
47 views

Recursive function definition, how does my teacher arrive at this answer?

I am currently revising for my maths exam in school and there is a section on recursion. The question is explained as follows: $$f(m, n) =\begin{cases} n + 1 &\text{if } m = 0\\ f(m − 1, 1) ...
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1answer
65 views

How many words can you make with {1,2,3,4} such that difference between 2 letters is 1

Hello and happy new year. I've been struggling with this question and I really need some help. The question is, find a recurrence relation that demonstrates how many words of length $n$ can we write ...
2
votes
5answers
60 views

stuck trying to solve nonhomogeneous recurrence relation

hello and happy new year! I'm trying to solve this question: We are required to find the solution (direct formula) of the following recurrence relation: $b(n)=b(n-1)+n-1$, $b(0)=0$. What I did: I ...
2
votes
2answers
82 views

Does there exist an infinite sequence $p_0,p_1,p_2…$ of prime numbers such that $p_k=4 p_{k-1}\pm 1$

$k \in Z^+$ firstly we know that there exists infinetly many primes of the form $4n+1$ by FTA also we see that if we consider finite primes say to $n$ then the recursive formular can be expressed ...
0
votes
1answer
66 views

Solving recurrences with summation factors (Concrete Mathematics)

Chapter 2 in Concrete Mathematics talks about solving recurrences of the form $$a_{n}T_{n}=b_{n}T_{n-1}+c_{n}$$ by reducing them into a sum. The authors multiply both sides by a summation factor ...
3
votes
2answers
83 views

Recurrence equation for $(-1)^k k$

In a project of mine I came across the recurrence relation $$ a_{n+1} = 1 -(n+1)\sum_{k=1}^n{\frac{a_k}{n-k+1}\binom{n}{k}},\quad a_1=2; $$ From calculating the first few terms it seems obvious that ...