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

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46 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
2
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31 views

How to solve a first order inhomogenous recurrence relation?

I have a recurrence relation for a fund that starts a 50 million, 6 % interest every year, and an outtake of 2 million/year. How to find out a solution for what funds exists after n years? ...
2
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0answers
31 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
2
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0answers
48 views

Recurrence Relation with $n^{\frac{1}{2^k}}$ Square Root

I tried to solve this recurrence relation, like the following: $$ \\ T(n) = \begin{cases} T(\sqrt[2]{n}) + c, & \text{if }n \geq 2 \\ a, & \text{otherwise} \end{cases} \\\\ T(n) = ...
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40 views

Verify that this recurrence relation is in O(log n)

For the recurrence $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ is in $\Theta(lg n)$ My attempt at a solution (mostly just wanting to verify its correct). Lower Bound: $T(n)=2*\lceil\frac{n+1}{2}\rceil+c$ ...
2
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101 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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139 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 ...
2
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861 views

Recurrence relation question

A country uses as currency coins with values of 1 peso, 2pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos. a) Find a recurrence relation for ...
2
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256 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 ...
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36 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 ...
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102 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
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79 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
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111 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
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72 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 $$a_{n}=L_{n}\left(\dfrac{17}{n}\right)$$ ...
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53 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 ...
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80 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 ...
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129 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 ...
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104 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 ...
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68 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 ...
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55 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 $\left[\sum_{k=0}^{m} P_k(n)f(n+k)\right]+P_{-1}(n)=0$. ...
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722 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 ...
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316 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 ...
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0answers
279 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: $a_0 = \alpha$; $a_1 = \beta$; ...
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0answers
188 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 ...
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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
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492 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 ...
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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 = ...
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279 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
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346 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 ...
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0answers
100 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 ...
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14 views

Recurrence Relation / Difference Equation Problem

I am trying to solve the following recurrence relation, but I am doing something wrong all the time when trying to find the particular solution, and I cannot figure out what. ...
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0answers
21 views

Complexity of $T(n) = T(n-10) + \sqrt{n}$

I'm using the iteration method to find the complexity of the following recurrence (I can't use the master theorem because it doesn't match the MT form). $$ T(n) = T(n-10) + \sqrt{n} \text{ and } T(1) ...
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16 views

Deriving binomial distribution from a recurrence.

Let $X_n, n\geqslant1$ be iid random variables with distribution $\mathbb P(X_1=1)=p = 1 - \mathbb P(X_1=0)$. Let $S_0=0$ and $S_n=\sum_{i=1}^n X_i$, $n\geqslant1$. Let $q_{n,k}=\mathbb P(S_n=k)$, for ...
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24 views

Sequence from generating function.

Consider the recurrence $$\mu_1=1, \mu_2=2, \mu_3=4, \mu_4=8, \mu_5=16, \mu_6=32 $$ and $$\mu_{n+6} = \mu_n + \mu_{n+1} + \mu_{n+2} + \mu_{n+3} + \mu_{n+4} + \mu_{n+5}, n\geqslant 1. $$ The generating ...
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27 views

Solving a difference equation with several parameters

Let $r>4$ be a positive integer. Let us consider this difference equation: $$u_{q+1}=(r^{2q+1}+(c/a))u_{q}-(c/a)r^{2q-1}u_{q-1} +2c+d-(bc/a)$$ where $a,b,c,d$ are integers. I want to find a ...
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24 views

Did i multiply the sums correctly?

This is an extention to this question except i am unsure of whether i have done it correctly: $$ y'' = -y'(f(x) - r(x) y') $$ $f(x) = \sum_{n=0}^\infty s_n x^n$, $y = \sum_{n=0}^\infty a_n x^n$, and ...
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0answers
30 views

solve third order scalar using 3 by 3 matrix recursion

Suppose I have something like the following: $$t(n+2) = 3t(n+1) - 2t(n) + t(n-1)$$ I do not want a complete solution to this question. All that I would like to know is how to convert this into a 3 ...
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20 views

Did i generalize this series solution correctly?

$$ f(x) = \frac{y''}{p(x)y'} + r(x) y' $$ if all functions are expressed in their power series form, then: $$ y = \sum_{n=0}^\infty a_nx^n $$ $$ p(x) = \sum_{n=0}^\infty p_n x^n $$ $$ r(x) = ...
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0answers
28 views

How do I solve this recurrence relation and prove by induction?

I have this summation formula: $T(n)=\sum_{i = 1}^{n}T(n-i)T(i-1)$. Base case is $T(0)=1$, $T(1)=1$. How do I find the recurrence relation and prove it by induction?
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12 views

Convergence of a recurrence relation

Some background information. I'm trying to estimate four probabilities $(p_1, p_2, p_3, p_4)$ (sum is equal to one) from nine numbers $(n_1, \dots, n_9)$ using Maximum Likelihood. (All $n_i$'s are ...
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36 views

Coefficients of the polynomials generated by $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$.

Let $f_0=x,\ f_{i+1}=f_i\pm\dfrac1{f_i}$ for $i\geq0$, i.e., $f_i=\dfrac{\sqrt{f_{i+1}^2\mp4}+f_{i+1}}2$ I have observed that $f_1=\dfrac{x^2\pm1}x$ $f_2=\dfrac{x^4\pm3x^2+1}{x(x^2\pm1)}$ ...
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24 views

Interesting recurrence equation models?

So I've had an introductory subject to recurrence equations and now I have to study a model using recurrence equations as a project. The problem is that most models I find are either elemental or ...
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18 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 ...
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39 views

2nd order difference equation with n dependent coeff

I wanted to know if there was and solution to the following equation. $\left(N-n+1\right)E_{n+2} - NE_{n+1} + \left(n+1\right)E_{n} + N = 0$ Where $E_0 = 0$ and $E_1 = 2^N - 1$. $N$ is just a ...
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74 views

$T(n) = T(n/2 - \log(n)) +1$ using Substitution Method

I have the following recurrence: $$T(n) = T(n/2 - \log(n)) +1$$ How can this be solved using the substitution method? I don't fully understand the theory of this method and I'm not sure how to apply ...
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0answers
55 views

Solution of $x_{k+1} = x_{k} (a x_{k} + b)$

Could anyone help me to solve the equation $x_{k+1} = x_{k} (a x_{k} + b)$, for find the explicit solution of $x_{k}$? BTW. Do you know a GOOD book for the classification for non linear difference ...
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0answers
26 views

Calculate the total cost

According to my notes: $$T(n)=T\left(\frac{n}{2} \right)+T\left(\frac{n}{4} \right)+T\left(\frac{n}{8} \right)+n$$ Th recursion tree is this: Cost per level $i \to \left( \frac{7}{8} ...
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0answers
26 views

Recurrence with Polynomial Coefficients of $n$

How would I solve a recurrence relation like $$a_n = f_1(n)a_{n-1} + f_2(n)a_{n-2} + ... + f_k(n)a_{n-k} + g(n)$$ where $f_1, f_2, ..., f_k, g$ are polynomials and $\left \{a_n\right \}$ is my ...
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0answers
58 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} ...
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0answers
26 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 ...