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

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1answer
20 views

Is it possible that a randomized recursion has a nonzero probability of either converging or diverging?

I have very little "hands-on" experience with probability, but here is my context: I was looking at the random Fibonacci sequence: $$f_0=f_1=1, f_n=f_{n-1}+Xf_{n-2}$$ where $X$ is chosen randomly ...
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0answers
20 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 ...
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17 views

Concrete math generalized josephus recursion understanding 1.15

I am studying through the josephus problem in concrete math , Here is the equation of binary form $$f(1) = α ;$$ $$f(2n + j) = 2f(n) + β_j ,$$ $$\text{ for } j = 0, 1 \text{ and } n \geq 1$$ this ...
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3answers
34 views

Why, intuitively, does the solution to a general linear recurrence relation make sense?

I reasoned through the solution to a differential equation, and $e^{\alpha x}$, for better or worse, seems to make sense. Each derivative sending the function to itself seems to suggest $e^{\alpha ...
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1answer
94 views

Does this sequence consist of squares of integers?

Question: let sequence $\{x_{n}\}$ such $$x_{0}=0,x_{1}=1,x_{2}=0,x_{3}=1$$ and such $$x_{n+3}=\dfrac{(n^2+n+1)(n+1)}{n}x_{n+2}+(n^2+n+1)x_{n+1}-\dfrac{n+1}{n}x_{n}$$ show that:$ ...
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1answer
30 views

Evaluation of a Hankel-like determinant

I consider the following determinant (Hankel-like?) $$ [f_1,f_2,...,f_n]:=\begin{vmatrix} f_1 & f_2 & \cdots & f_{n-1} & f_n\\ n-1 & f_1 & \cdots & f_{n-2}& f_{n-1}\\ 0 ...
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1answer
48 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
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2answers
102 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 ...
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1answer
34 views

Prove that the elements of these two sequences are not null

Let $x_{n+1}=x_n+2y_n$ and $y_{n+1}=y_n-x_n$, where $x_1=1$ and $y_1=-1$. I tried proving by contradiction, I tried by induction, I got nothing. This is a question I had on an exam, I didn't manage ...
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1answer
24 views

Recurrence relation and combinatorics

I am reading p.4 of the article http://mercercountymathcircle.files.wordpress.com/2014/03/recurrence_relations.pdf which consider the following problem: Find the units digit of ...
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3answers
295 views

Homework - Resolve the recurrence relation

What's the closed formula of this recurrence relation? $$a_n = a_{n-1}+2a_{n-2}+2^n \text{ with } a_0=1, a_1=2 $$
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0answers
32 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
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0answers
26 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 ...
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0answers
14 views

Comparing Generalized Continued Fractions

Gosper lays out a method (under Approximations) for comparing regular (a.k.a simple) continued fractions which have all partial numerators set to 1. Continue comparing terms until they differ, then ...
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1answer
60 views

Solving this recurrence relation

please tell me a way to solve this recurrence $$n\cdot R_n=C_1\cdot R_{n-1}+C_2\cdot R_{n-2}.$$ $C_1$ and $C_2$ are constants.. There is an $n$ there in the left hand side.. it makes mess. I tried ...
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0answers
57 views

solution of a recurence equation

The equation is $f(n)=\left(1+\frac{na}{b}\right)f(n-1)-\frac{na}{b}f(n-2)-\frac{n}{b}$ where $a,b>0$ and real, and $f(0)=0$, $f(1)=c\in R$. I expressed $f(2)$, $f(3)$, $f(4)$ and $f(5)$ in ...
3
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1answer
28 views

Newton Rhapson Algorithm Accuracy

I read somewhere that the NR algorithm in general (given an appropriate initial value) increases in accuracy by roughly two decimal places per iteration. Is this something that can be proven, or is ...
6
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0answers
84 views

Nonlinear inhomogeneous recurrence $f(x)^2=f(x+1)+S(x)$ to find nested radical

I want to solve the recurrence relation $$f(x)^2=f(x+1)+S(x)$$ where $S(x)$ is a given polynomial. The background is to find nested radicals expressions of the form ...
4
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1answer
91 views

Going from closed form to recurrence relation

If I had a closed form for a sequence that I suspect to represents a recurrence relation how would I determine the recurrence relation? In particular, I have the sequence $$a_n = ...
0
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1answer
34 views

Recursive Relationship: Paper Towels

A standard roll of paper towels consists of a cardboard tube with outer diameter $4$ cm. Imagine the paper being wound onto the cardboard tube. After each complete winding, the total diameter of the ...
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8answers
254 views

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

Question: Proof that if $$a_n=\left\{ \begin{array}{ll} a_1=1\\ a_{n+1}=1+\frac{1}{1+a_n} \end{array} \right.$$ then $a_n$ converge, and then find $\lim_{n \to \infty}a_n$. I found ...
27
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3answers
492 views

Finding every $n$ such that $a_n$ is an integer

Let us define $\{a_n\}$ as $a_1=a_2=1$,$$a_{n+2}=a_{n+1}+\frac{a_n}{2}\ \ (n=1,2,\cdots).$$ Then, is the following true? If $a_n$ is an integer, then $n\le 8$. I conjectured this by using ...
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2answers
43 views

recurrent events-Probability of even number of successes

Let E be the event of an even number of successes. $u_n$:Probability of E occurring at the nth trial not necessarily for the first time $f_n$:Probability of E occurring at the nth trial for the first ...
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0answers
16 views

Produce a list of the most-similar units, given various correlations/relationships

I have a database full of units (U1 - U50, U51...) where every unit has the same standard attributes (A1 - A10) and where a % of each attribute defines the amount of that attribute for that particular ...
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0answers
39 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) + ...
2
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1answer
54 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
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4answers
339 views

Converting recursive equations into matrices

How do we convert recursive equations into matrix forms? For instance, consider this recursive equation(Fibonacci Series): $$F_n = F_{n-1} + F_{n-2}$$ And it comes out to be that the following that ...
3
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2answers
50 views

Recurrence Master Theorem Question with asymptotic Upper and Lower Bounds

If I were to solve the recurrence of following equation and give asymptotic upper and lower bounds: $$T(n) = 4T(\frac{n}{2}) + n^2 + n$$ Can I apply Master Theorem on this? My attempt was following: ...
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0answers
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Queuing theory-Multiple server (reducing simple recurrence formulas)

The equations given in 6.3 have been reduced which really eases the computation in further studies. But I tried to find the method of reducing these but I could not find a way at all. Any hints will ...
2
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2answers
64 views

Finding a Closed Form for a Recurrence Relation

I know that a general technique for finding a closed formula for a recurrence relation would be to set them as coefficients of a power series (i.e. a generating function). Then use properties of ...
1
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0answers
18 views

non-homogenous recurrence relation, with split boundary conditions

I have non-homogenous recurrence relation: $x_{t+1}=\alpha x_t+\beta x_{t-1}+\gamma$ with the following boundary conditions: $x_2=\alpha x_1+\gamma$ $x_{T}=1/2x_{T-1} +1/2$ Anyone know how to ...
3
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1answer
37 views

What's the time complexity of T(n)=nlogn+T(n-1)?

The title says it all. The best I can come up with is that this expands to T(0) + 1log 1 + 2log 2 + ... + (n-1)log (n - 1) + nlog n which is ...
3
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1answer
70 views

Limit of a recursiv trigonometric function

So while doing my math homework I recently stumbled upon this little thing: It seems that $y_n = \sin(y_{n-1}) + k$ with arbitrary $y_0$ and $k$ converges to a definite value for $n \to \infty $. ...
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44 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
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1answer
67 views

Fibonacci numbers $F_{n+3} + F_{n} = 2F_{n+2}$

Prove $F_{n+3} + F_{n} = 2F_{n+2}$ for any positive integer n. So What I did was this: fn+ fn+1 = fn+2 fn + fn+1 = fn+2 => fn+2 -fn+1 fn+1 + fn+2 = fn+3 then I subsituted into equation in ...
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1answer
70 views

Help finding the closed formula for a recurrent relation

In the last steps of finding the complete solution of a linear differential equation by a power series, I got stuck on finding the closed formula for the following recurrent relation: $$B_n = B_{n-1} ...
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0answers
69 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: ...
3
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1answer
104 views

General Solution of Functional Equation

What is the general solution to: $$ \frac{f(x+h)-f(x)}{h} = \frac{1}{x} \tag{1} $$ Obviously the solution to this for the limiting case of $h\to 0$ is $f(x) = \ln(x) + c$ Attempting to solve the ...
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2answers
38 views

Generating function satisfying a second degree equation

I got this problem in an exercise list: Let $G(x)$ be the generating function of the numeric sequence $(C_n; n \geq 0)$ satisfying the recurrence equation: $$C_n = \sum_{k=0}^{n-1}C_kC_{n-k-1}, ...
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0answers
30 views

Standard results for limit of recurrence relation?

Is there a standard result out there that gives the following? It looks graphically like it must be true, but I'd like to appeal to a known result if possible. Let $f:[0,1]\to[0,1]$ be a continuous, ...
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0answers
19 views

Solving recurrence relation with $c=\Theta(\log(\log(n)))$

The master theorem in recurrence relations, i encounter with difficulty in solving problems, which have $c=\Theta $ $($of something$)$ or when it has $\log(n)$ or its variants instead of a simple ...
2
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1answer
70 views

Quadradic recurrence relation

There is an method to solve recurrences of the form $a_{n+1} = (a_n + c)^2$? I am particularly interested when $c = 1$. I tried to use generating functions but I got stuck with. Let $G(x) = \sum_{k ...
3
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1answer
85 views

Solving a recurrence relation ${}$

I feel I'm wasting my time trying to solve this $a_0$ is given $\displaystyle a_{n+1}=\frac{n-1}{n+2}(a_n-n-2)$ Mathematica found a closed form but there's a problem when evaluating for ...
3
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2answers
99 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...
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1answer
62 views

Divide and Conquer (recurrence relation problem)…

The problem: (a) Use a divide-and-conquer approach to devise a procedure to find the largest and next-to-largest numbers in a set of n distinct integers. (b) Give a recurrence relation for ...
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1answer
23 views

Change of variables in function $T(n)$.

I've been given this recurrence to solve: $T(n) = T(\sqrt n) + \theta(lglgn)$ And I'm told that the way to solve it is to let $m = lgn$, so that the recurrence can be rewritten as follows: $S(m) = ...
6
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1answer
124 views

Limit related to the recursion $a_{n+1}=(1-n^{-1/4}a_n)a_n$

Let $0<a_{1}<1$, with $$a_{n+1}=a_{n}(1-n^{-\frac{1}{4}}a_{n})$$ Does there a real numbers $A$ which make the limit $$ ...
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3answers
46 views

Recurrence relations for students of the third year of secondary school.

I am not able to solve this problem in order to find a explicit form for the recurrence relation (note: in the original text I can read "a with n" and "a with n-1", but I am not able to format here) ...
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2answers
92 views

How to solve the recurrence $T(n) = T(n-1) +\sqrt{n}$?

While solving the recurrence of the title I come to the series $$T(n) = \sqrt1 + \sqrt2 + \sqrt{3} + \cdots + \sqrt n.$$ Please somebody help me how to solve this.
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0answers
64 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...