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

learn more… | top users | synonyms (2)

2
votes
0answers
68 views

(limit of) a linear second order recurrence relation with variable coefficients

I have the following recurrence relation: $(n + 1) a_{n + 2} = (w (n + 1) - c) a_{n + 1} - z (n + 1)*a_{n}$ that I would like to either solve, or to get the $n$ goes to Infinity limit of the ratio ...
2
votes
0answers
252 views

Induction proof of a recurrence relation

I have some trouble with an induction proof for the following problem. There is a vending machine that only takes coins of value 1 and 5 respectively. Let $S_n$ be the number of different ...
2
votes
0answers
33 views

Determinant of Stirling submatrix

Consider the table containing the Stirling numbers of the second kind $S\left( n,k\right) $. From this table construct a square matrix $M$ containing $N$ differents rows from the table ...
2
votes
0answers
95 views

How to solve this recurrence relation (related to discrete Fourier transform)?

I am having trouble with the following recurrence relation: $$c_{n+1} - c_{n-1} = 2\alpha \sin \frac{(2n-1)\pi}{N} c_n, \quad\forall n \in \mathbb{Z},$$ where $N$ is odd and the initial condition is ...
2
votes
0answers
76 views

Recurrence relation for polygamma reflection polynomials

In the reflection formula for the polygamma function $$ \psi^{(n)}(1-z) + (-1)^{n+1}\psi^{(n)}(z) % = (-1)^{n} \pi \frac{d^{n}}{d z^{n}} \cot (\pi z) $$ the right hand side is a polynomial ...
2
votes
0answers
102 views

Showing the relationship between Lucas and Fibonacci numbers with a recursive relationship

Given the equation $$L_{n+k}-(-1)^kL_{n-k}=5F_kF_n$$ Where $\{L\}$ is the set of Lucas numbers and $\{F\}$ is a Fibonacci sequence, $n$ and $k$ are both $>0$ and integers, how would one develop ...
2
votes
0answers
71 views

How prove this sequences have limts and find this value?

let $f:R\longrightarrow R$,and $f(x)=x-\dfrac{x^2}{2}$,and $$L_{1}(x)=f(x),L_{2}(x)=f(f(x)),L_{3}(x)=f(L_{2}(x)),\cdots,L_{n}(x)=f(L_{n-1}(x))$$ and let $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
2
votes
0answers
52 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
2
votes
0answers
76 views

Solving recurrence relation of algorithm complexity?

Supposing I write an algorithm that results into this kind of recurrence relation $$\left\{ \begin{array}{ll} T(0)=T(1)=1 \\ T(n)=T\left(\lfloor n/2 \rfloor \right)+T\left(\lceil n/2 ...
2
votes
0answers
123 views

Recursive Sequence from Finite Sequences

I'm searching for the name of these kind of sequences: Suppose you start off with a finite sequence containing one term: S0 = {3} To get the next sequence you ...
2
votes
0answers
103 views

Is there a closed form for this recurrence?

Given $$ E_{n,k} = \begin{cases} 0 & \text{ if } n \leq k \\ n & \text{ if } k = 0 \\ \sum_{i=0}^{n-1} \dfrac{1}{n} \cdot E_{i,k-1} & \text{ otherwise } \end{cases} $$ I wonder is there ...
2
votes
0answers
67 views

How much information do I gain from each modular inequality?

Problem details: Let $p = 2^{48}$, $s\leftarrow\{0,1\}^{48}$, and $a,b$ known constants. Furthermore let $f(x) = a x + b \pmod{p}$ and let the value $r_k$ be defined by the first-order recurrence ...
2
votes
0answers
54 views

how to compute $\lim_{n\to\infty}\ln(f(n))/\ln(n)$ for this given $f(n)$?

Let $c=f(0)$ be a real number and $n,m$ positive integers. Let $P_k(n)$ be integer polynomials of $n$. Let $f(n)$ be defined by the recursion $\left[\sum_{k=0}^{m} P_k(n)f(n+k)\right]+P_{-1}(n)=0$. ...
2
votes
0answers
657 views

Probability of tossing a biased coin without having k heads consecutively in a row

I got asked by a friend this question; I have a coin, the probability of receiving a head by tossing is $p$ and tail $1-p$. I have to toss it $n$ times without getting $k$ heads in a row. What is the ...
2
votes
0answers
293 views

Linear multivariate recurrences with constant coefficients

In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any ...
2
votes
0answers
252 views

Solving two-parameter linear recurrence with different initial values

I'm looking for a method to solve the well-known recurrence ralation with different initial values. The one-parameter linear recurrence is easier. Let's take a peek: $a_0 = \alpha$; $a_1 = \beta$; ...
2
votes
0answers
181 views

How to solve the recursive relation in Kalman filter?

I was wondering how to solve the Kalman filter's recursive equation (also see the appendix at the end of this post) for the estimated state $\hat{\textbf{x}}_{n|n}$ at time $n$, over discrete times ...
2
votes
0answers
75 views

Construction of polynomials with non-commutative elements.

I have a simple set of polynomials which I know how to construct for each integer $n$, but I havn't been able to write them down in terms of concrete sums and products. For $n\in\mathbb N_+$, we have ...
2
votes
0answers
483 views

recurrence relation with non constant coefficients

I'm trying to solve a second order differential equation and I got a recurrence. Can someone help to solve $$n(n-1+q)a_{n}-a_{n-3}+e\cdot a_{n-2}=0$$ where $q$, $e$, and $a_{0}$ are some real numbers ...
2
votes
0answers
69 views

On bounding the average cost of top-down merge sort

Let $A_n$ be the average number of comparisons to sort $n$ keys by merging them in a top-down fashion (see any algorithm textbook). It can he shown that $$ A_0 = A_1 = 0;\quad A_n = ...
2
votes
0answers
269 views

Bell-like recurrence

Let $$A(n)=\sum_{k=0}^{n-1}\binom{n}{k}A(k)+n!,\quad A(0)=1$$ $$B(n)=\sum_{k=0}^{n-1}\binom{n}{k}B(k)-n!-n!\sum_{k=1}^{n}\frac{1}{k!},\quad B(0)=-1.$$ I'm interested in computing $S(n)=A(n)+B(n)$ ...
2
votes
0answers
329 views

Gamblers ruin difference equation

I'm not sure where my mistake is in the following. Gambler starting with k dollars and playing a $50/50$ game where he increases in wealth by one dollar or decreases by one dollar until achieving $N ...
2
votes
0answers
98 views

Systems of partial difference equations

I would like to ask, if any general method of solving systems of partial difference (not differential) equations is known, at least for some classes of systems. For instance, for ordinary difference ...
1
vote
0answers
10 views

Asymptotic Notations Iterative Method for Solving Recurrences

Recurrence T(n)= T(n^1\2) + O(lg(lg(n))) The solution suggests substituting m = lg(n) So the recurrence becomes S(m)= S(m\2) + O(lg(lg(m))) Then solving using iterative method for solvng ...
1
vote
0answers
39 views

How to solve this tough recurrence relation?

For solving a related probability problem, I'm required to solve the following recurrence relation:- $q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times ...
1
vote
0answers
13 views

Finding Recurrence Relation in Closed Form or as Infinite Summation/Coproduct

My recurrence relation is: $$f(n)=(2n-3) f(n-1) -x^2 f(n-2)$$ Where $$f(-1)=1$$ $$f(0)=0$$ $$f(1)=-x^2$$ $$f(2)=f(1)$$ $$f(3)=-x^2 (3-x^2)$$ And it gets more complicated from there
1
vote
0answers
49 views

Trace and transpose of a Matrix

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1 =sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2} =\frac{s}{n+2}\{ R_{n+1} ...
1
vote
0answers
24 views

Solution of ODE with initial values

Given data in the problem ${\psi'(t)}_{3 \times 3}=A_{3 \times 3}\psi(t)_{3 \times 3}, \psi(0)_{3 \times 3}=R^{cl}_{3 \times 3} \\ \phi'(t)_{3 \times 3}=t\hspace{.1cm}B_{3 \times 3} \phi(t)_{3 ...
1
vote
0answers
34 views

Recurrence relation sequence (Hard)

I was wondering if this sequence was possible using two base cases? The sequence is $\{-1,0,1,3,13\}$ .. Ive came close by doing $(a_{n-1})^2+a_{n-2}+a_n-1$ which works for everything but the $1$. I'm ...
1
vote
0answers
42 views

Difference equation of second order system with zero

I saw from lecture notes that difference equation of a first order system is like this: (1) (2) (3) (4) 1. What happens between (3) and (4)? It looks like inverse Z-transform but according ...
1
vote
0answers
14 views

Complexities of Recurrences in the form $t(n) = t(\alpha n) + t(\beta n) + cn$

When considering recurrence relations, they are generally of a form similar to $$t(n) = t(\alpha n) + t(\beta n) + cn$$ and there are three cases to be considered for our values of $\alpha$ and ...
1
vote
0answers
33 views

Recurrence Derivative

I have a recurrence relation as follows $ \left\{ \begin{array}{ll} R_0=H & \mbox{if } n = 0 \\ R_1(s)=sR_0 \hspace{.1cm} A & \mbox{if }n=1\\ R_{n+2}(s)=\frac{s}{n+2}\{ ...
1
vote
0answers
30 views

What does “combining the solutions in O(n) time” mean?

Algorithm $X$ proceeds by recursively solving $5$ subproblems of one-half the size, then combining the solutions in $O(n\log n)$ time. Algorithm $Y$ makes $9$ recursively calls on ...
1
vote
0answers
68 views

What is the recurrence relation between $a_n, a_{n-1}$ , $a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx$

I would appreciate if somebody could help me with the following problem Q: What is the recurrence relation between $a_n, a_{n-1}$ ? $$ a_n = \int_0^1 {x^n}\tan\left( \frac{\pi}{4}x\right) dx,\ \ ...
1
vote
0answers
39 views

Non-recursive closed-form of the coefficients of Taylor series of the reciprocal gamma function

The reciprocal gamma function has the following Taylor series. $$\frac{1}{\Gamma(z)}=\sum_{k=1}^{\infty}a_kz^k,$$ where the $a_k$ coefficient are given by the followint recursion. $a_1=1$, ...
1
vote
0answers
32 views

Complicated 2-variable recurrence

I am analyzing some kind of construction for which size satisfies the following recursion: $P(n,k)=\frac{k}{2}\log \frac{k}{2} + 1 + \frac{n}{2} + P(\frac{n}{2},k) + P(\frac{n}{2},\frac{k}{2})$ ...
1
vote
0answers
19 views

Solving a recurrence with a term $T(\frac n 2 + 2)$

I'm stuck trying to solve the following recurrence: $$\begin{align*} T(n) &= 4T(\frac n 2 + 2) + n : n > 8\\ T(n) &= 1 : n \leq 8 \end{align*}$$ In particular, I'm not sure how to deal ...
1
vote
0answers
32 views

Need help checking my recurrence for a simple algorithm

All I'm writing to get a second opinion on the algorithm shown in this link. I'm pretty sure its supposed to be $T(n)=2T(n/2)+n$ but I can't see where I'm supposed to get the +n from. So far I'm ...
1
vote
0answers
108 views

Method for deriving an Exponential Moving Average?

I have a formula for an exponentially weighted moving average function defined recursively as: $S_t = a*Y_t+(1-a)*S_{t-1}$ Where: $a\in (0,1)\cap \mathbb{Q}$ $t$ represents time $Y_t$ is the value ...
1
vote
0answers
42 views

Using a Recursion Tree to solve the recurrence $T(n) = \sqrt n T(\frac{n}{2}) + 10n$?

I am attempting to solve the above recurence by giving tight $\Theta$ bounds. Assume that the logs here are all base 2! To solve a recursion tree as far as I understand, I need two things. The ...
1
vote
0answers
12 views

Is there a way to express a closed form for a partial derivative of this recurrence relation?

Here's the relation: if $n\ge j:$ then $$ \sigma(n,j,d) = d \cdot\left( \log j-\sigma\left(\frac{n}{j}, 1+d, d \right)\right)+\sigma(n, j+d, d )$$ And here's the terminating condition if $n < j$ ...
1
vote
0answers
40 views

Finding Function Representation of Recursive Sequence

I was trying to find one of the roots of $x^2 + 4x + 3 = 0$ by deriving a continued fraction from the recursive formula $x = -3/x - 4$ (every step of the approximation you increase the recursion by ...
1
vote
0answers
62 views

Explicit formula for inverse function for n-th partial sum of harmonic series

Originally, I was trying to work out an explicit formula for a function with the property that $f(x + \frac{1}{f(x)}) = f(x) + k$ for some $k$. This is inspired by a thing I did with a metronome ...
1
vote
0answers
29 views

Asymptotic behaviour of two dependent recursive sequences

I have two sequences whose relation is described in the following recurrence relations: $ p_{k + 1} = p_k + \frac{1}{2s_k}$ $ s_{k + 1} = s_k + \frac{s_k}{p_{k+1}}$ (when $p_0=2, s_0 = ...
1
vote
0answers
51 views

Recurrence relation for Binary String Question

I have a question which has been a little stumped. I'm pretty sure I know the answer, but don't know how to prove it to be true. Here it is: "Given an infinite length random binary string, what is ...
1
vote
0answers
33 views

Question about finite sums and integer recursions.

Let $n$ be a positive integer and let $g(n)$ be a given strictly increasing integer function such that $0<g(n)<n$ for all $n$. Also the sequence $ |g(n) - n|$ is unbounded as $n$ grows. Let ...
1
vote
0answers
42 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
1
vote
0answers
33 views

Possible to determine if a more 'compact' solution to a linear recurrence exists?

Given a linear a recurrence relation. It is possible to express a solution in terms of summations, products, and the coefficients which appear in the recurrence. For example, in the case of a single ...
1
vote
0answers
45 views

General Solution to Linear Difference Equation: Is this correct

Notation: Let $$Q_1[f(x)] = \lim _{h \rightarrow 1} \frac{f(x + h) - f(x) }{h}$$ And let $$Q_1^{-1}\left[f(x)\right] = G(x)|Q_1\left[g(x)\right] = f(x)$$ Consider the equation $$a_0(x) + ...
1
vote
0answers
74 views

Need help with these recurrence relations

I had received some challenging recurrence last week, I did most of them except this and also one of its kind. It states Given $a_0=0$ and $a_1=1$, solve these recurrence relations: ...