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

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1answer
65 views

Prove that the recurrence is true [duplicate]

I am working on an assignment question, and am having trouble moving ahead. The question is as follows: Let the total number of bit strings with three consecutive zeros be $t_n$. Prove for $n \ge ...
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2answers
94 views

Recurrence relation of order $n$: $f(n) = \dfrac{1}{k-1}\sum\limits_{i=1}^n {n \choose i} f(n-i)$.

I came across this recurrence relation while looking for a closed form for $S(n,k) = \sum\limits_{i=0}^\infty \dfrac{i^n}{k^i}$.After a few manipulations, I came across this recurrence relation: $f(n) ...
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1answer
122 views

Using induction to prove a general form from a recurrence relation

I have the recurrence relation: $a_{n+2} = \dfrac{-a_{n}}{(n+2)(n+1)}$. I have done some work to identify that two cases emerge: one is n is positive, and one if n is negative. If n = 2m (even) ...
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1answer
63 views

Creating Recurrence

If I have an integer $n \geq1$, and I had to draw $n$ straight lines, so that no two of them are parallel as well as no three of them intersect in one single point. These lines divide the plane into ...
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1answer
54 views

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?
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1answer
406 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
45 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
93 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 $a_1=2$? $$$$ ...
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0answers
103 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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1answer
27 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
42 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
34 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
52 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.
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1answer
40 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 ...
3
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3answers
183 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: $a_{n+1} = a_n - 0.069\cdot b_n$ $b_{n+1} = b_n + 0.069\cdot a_{n+1}$ ...
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1answer
201 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 ...
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1answer
284 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
117 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 ...
3
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1answer
195 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, ...
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3answers
96 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
99 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
66 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
57 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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0answers
95 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
160 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
46 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 ...
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3answers
49 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 ...
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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 ...
3
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1answer
27 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
42 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
175 views

How to make recursive computation of $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ stable?

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 the integral $I_n=\int_0^1 \frac{x^n}{x+\alpha}\,dx$ ...
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2answers
111 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
387 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
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2answers
60 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
246 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: ...
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0answers
25 views

Difference equation - counting problem

I need to to define difference equation for following problem and solve that equation using generating function. Border of length 10cm is made of small bricks (10cm long) and large bricks (20cm ...
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2answers
49 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 ...
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3answers
166 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 ...
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1answer
710 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 ...
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1answer
35 views

Difference equation formula $\sum a^t = \frac{a^t}{a-1}$.

As I explain below, this question was originally posted by user YYG, but then deleted. I am reposting the question (from memory) and I will answer it myself below. Question: In Difference Equations ...
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0answers
140 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
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1answer
78 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
136 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
80 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
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1answer
114 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
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1answer
135 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} ...
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5answers
133 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
73 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) ...
0
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1answer
90 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 ...