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

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Combinatorial explanation to following recurrence relation $a_n = 2 a_{n-1} + a_{n-2}$

Question was the following: $a_n$ is the number of ternary strings (strings of 0,1,2) which contain no consecutive zeros and no consecutive ones. Find a formula for $a_n$? By brute force, I found a ...
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4answers
5k views

What is the solution to the following recurrence relation with square root: $T(n)=T (\sqrt{n}) + 1$?

This looks like a question asked earlier, but it isn't. $$T(n)=\begin{cases} T (\sqrt{n}) + 1 \quad & \text{ if } n>1 \\ 1 & \text{ if }n=1\end{cases}$$ My professor gave this to me in ...
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8k views

How to solve this recurrence $T(n) = 2T(n/2) + n\log n$

How can I solve the recurrence relation $T(n) = 2T(n/2) + n\log n$? It almost matches the Master Theorem except for the $n\log n$ part.
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1answer
342 views

Expanding the generating function of the Fibonacci numbers to find a cute formula

$F_0=1$, $F_1=1$, $F_n=F_{n-1}+F_{n-2}$. The generating function is $-\frac{1}{x^2+x-1}$. I have to expand it to prove that $F_n=\sum_k\binom{k}{n-k}$. Could you help me please?
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3answers
1k views

How to solve this recurrence relation

What are some ways to solve this recurrence relation: $a(n+1)=2 a(n) - a(n-1) -1, \text{ with }a(0)=0, a(10)=0?$ I tried to first convert this inhomogeneous equation into a homogeneous one following ...
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3answers
375 views

The number of length-n ternary sequences with even ones and even zeroes

Just starting to appreciate recurrence relations Let $T_n = $ number of length-n ternary sequences with an even number of ones and an even number of zeroes. $T_0 = 1$, because $0$ is an even number, ...
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2answers
114 views

Proving integrality of a sequence of numbers

How would one go about proving whether or not the terms of a sequence such as the following $$1, 1, 1, 1, 1, 6, 21, 181, 5221, 1090981, 986241401, 51490676208426,\ldots$$ defined by ...
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2answers
126 views

Why should we suspect that the recurrence $T(n) = T(n-1) + n(n-1)$ satisfies a polynomial identity?

In the question Algorithms: Recurence Relation, the author asked about the recurrence relation $$T(n) = T(n-1) + n(n-1)$$ and one of the answers proposed assuming $T(n)$ is polynomial, then ...
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4answers
1k views

Chain rule for discrete/finite calculus

In the context of discrete calculus, or calculus of finite differences, is there a theorem like the chain rule that can express the finite forward difference of a composition $∆(f\circ g)$ in ...
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1answer
191 views

Solving Recurrence $T_n = T_{n-1}*T_{n-2} + T_{n-3}$

I have a series of numbers called the Foo numbers, where $F_0 = 1, F_1=1, F_2 = 1 $ then the general equation looks like the following: $$ F_n = F_{n-1}(F_{n-2}) + F_{n-3} $$ So far I have got the ...
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1answer
685 views

On the generating function of the Fibonacci numbers

Let's define the Fibonacci numbers as $F_0=1$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$. Using this recurrence I was able to calculate the generating function of the Fibonacci numbers to be ...
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3answers
144 views

Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$

I'm looking for a solution $f$ to the difference equation $$f(i)=2(f(i-1)+f(\lceil i/2\rceil))$$ with $f(2)=4$. Very grateful for any ideas. PS. I've tried plotting the the initial values into ...
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4answers
138 views

Recurrence equation: $u_n = 4u_{n−1} + 4u_{n−2}$ ; is $4x+4 = 4$ the characteristic equation?

Given this recurrence equation: $u_1 = 0, u_2 = 1$ $u_n = 4u_{n−1} + 4u_{n−2}$ Is the correct characteristic equation: $4x+4 = 4$ EDIT: Complete solve: The characteristic equation is ...
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2answers
177 views

Solving a recurrence for a probability?

I came across the following recurrence relation when exploring properties of a certain type of randomized perfect binary tree: $$ T(0) = \frac{1}{2} $$ $$ T(k + 1) = 1 - T(k)^2 $$ (Specifically, ...
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1answer
191 views

Finding the general formula $a_n$ for $a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$

How to calculate the general formula $a_n$ for the following sequence: $$a_n = \frac{1}{2a_{n-1}} + 2a_{n-2}$$ where $a_1=\frac{1}{2}, a_2=\frac{1}{4}$
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436 views

Whats better: 1 million dollars in a month or a penny(USD) doubled (and added) every day for 30 days?

THis is a question that I remember when I was in the 5th grade that tested our logical reasoning skills. And it is a simple choice knowing that the pennies doubling every day is an exponential ...
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1answer
360 views

Summation Of Product Of Fibonacci Numbers

Im trying to find out a general term for the following summation of products of fibonacci numbers:-- $$\sum_{k=4}^{n+1} F_{k}F_{n+5-k}\; , n \geq 3$$ I tried using Binet's equation but I am ...
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402 views

Understanding why the roots of homogeneous difference equation must be eigenvalues

There is some obvious relationship between the root solutions to a homogeneous difference equation (as a recurrence relation) and eigenvalues which I'm trying to see. I have read over the wiki article ...
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1answer
431 views

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is an exercise from Chapter 1 of "Concrete Mathematics". It concerns the Towers of Hanoi. Are there any starting and ending configurations of $n$ disks on three pegs that are more than $2^n - ...
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598 views

Solving recurrences of the form $T(n)=aT(n/a)+Θ(nlgn)$

On pages 95 and 96 of the third edition of the CLRS book, we find the following, which applies here since $a=b$ is all it takes to block the application of the Master Theorem: "Although $n\lg n$ is ...
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173 views

Recurrence relation: composition of a polynomial function

Let $f(x)=a+bx^2$. Define $f_n(x)$ to be the $n$-fold composition of $f$. That is $$f_1(x)=f(x)$$ $$f_2(x)=f \circ f(x)$$ $$f_n(x)=f \circ f_{n-1}(x), n \ge 2$$ Is there a way to find a formula for ...
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1answer
237 views

When is a recurrence relation linear

In http://www.cs.fsu.edu/~lacher/courses/COT5405/spring07/notes2.html, it says that $T(n) = aT(n/b) + f(n) $ is nonlinear recurrence. But I think it is linear because $T(n)$ is linear in $T(n/b)$. ...
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Second order homogeneous linear difference equation with variable coefficients

I was wondering if you would point me to a book where the theory of second order homogeneous linear difference equation with variable coefficients is discussed. I am having difficulties in getting ...
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3answers
749 views

how to solve $T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}$

How can I solve the recurrence $$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$ Please don't just write the name of the method, as I just started learning this stuff and things are a ...
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2answers
219 views

Method of solving no-homogeneous recurrence equation

I need to obtain a closed form of $M(t)$, satisfying the following recurrence equation: $$M(t+1)=a+bM(t)+\frac{c}{t+1}\sum_{t'=0}^tM(t')+df(t)$$ Where $f(t)$ is a known function and $a$, $b$, $c$ ...
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659 views

Number of Regions in the Plane defined by $n$ Zig-Zag Lines

Fellows of Math.SE, I have been scratching my head at a solution to an exercise in Donald Knuth's Concrete Math. Here is the problem: Here is the solution (I hid it in case someone wants to solve ...
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67 views

Divergence of $a_{n+1}=\sqrt{2a_n+3}$?

I am wondering what I am missing from my proof. I would like to show that the limit of the sequence $$a_{n+1}=\sqrt{2a_n+3},\,\, a_1=4,$$ goes to $\infty$, as $n \rightarrow \infty$. Is there ...
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2answers
179 views

Let $x_{n+1} = x_n + 1/(x_1 + x_2 +\ldots + x_n)$ with $x_1 = 1$. Show that $x_n\sim\sqrt{2\log(n)}$.

As the title states we have a sequence defined by $$x_{n+1} = x_n + \dfrac{1}{x_1 + x_2 + \cdots + x_n}$$ with $x_1 = 1$. The first few terms are: $1, 2, \frac{7}{3}, \frac{121}{48} \cdots$ Any ...
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Solve the recursion, $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$

Bring the following recursion relation to an explicit expression: $$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$$ $a_{0} = 0$, $a_1 = 1$, $a_2 = 2$ All the examples I have seen were with maximum 2 steps back ...
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2answers
64 views

Let $A$ be a matrix sized $p\times p$, where $2\le p$. Using recurrence relations, describe $A^k$.

Let $A$ be a matrix sized $p\times p$, Where $2\le p$. The matrix values in the main diagonal are $0$ and the rest are $1$'s. Example for $A$ where $p=5$: $$\begin{bmatrix} 0 & 1 & 1 & 1 ...
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1answer
301 views

two-dimensional recurrence

Can someone using only these conditions $$a_{m,k}=a_{m-1,k}+a_{m-1,k-1},m>k$$ $$a_{m,k}=1,m=k$$ $$a_{m,k}=0,m<k$$ prove that $$a_{m,k}=\frac{m!}{k!(m-k)!}$$ here is way to construct Pascal ...
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1answer
454 views

Show that the Catalan numbers are given by this recurrence relation

Hey guys! I'm doing an assignment, and I'm just not sure (at all) how to start this problem. Can somebody nudge/shove me in the right directions? Show that the Catalan numbers are given by the ...
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1answer
172 views

Complicated Multivariate Recurrence Relations For Generating Polynomials

I have the following multivariate recurrence relations all from the same system: First, suppose that $0\le k\le j\le m$, and let $N$ be an independent integer. Then we have for expressions $a(k,~ m,~ ...
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1answer
155 views

What is the expected number of questions answered to complete a sequence in which wrong answers send you to the start?

Given a sequence of n questions that each contain x answer choices, what is the expected number of questions answered before answering all questions correctly if answering a question incorrectly sends ...
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1answer
137 views

Period doubling bifurcation in a quadratic map

I am attempting the find $\mu$ for which the map $$x_{n+1} = \mu + x_n^2$$ undergoes a period doubling bifurcation. I understand that finding the fixed points of the map is the first step towards ...
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1answer
288 views

Recurrence for perfect matchings revisited.

I like to study combinatorics a bit as a hobby, and recently a question I found interesting was posed asking to derive a recurrence for the generating function $G_n(x)$, the ordinary generating ...
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2answers
96 views

Proving a solution to a double recurrence is exhaustive

The equation $$ b^2 = \frac{a(a+1)}{2} + 1 $$ where $a$ and $b$ are integers, has the following smallish-integer solutions: ...
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1answer
88 views

Expressing a recursively defined function in terms of factorials or gamma function

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = ...
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368 views

How to solve this recurrence $T(n)=2T(n/2)+n/\log n$

How can I solve the recurrence relation $$T(n)=2T\left(\frac n2\right)+\frac{n}{\log n}$$? I am stuck up after few steps.. I arrive till $$T(n) = 2^k T(1) + \sum_{i=0}^{\log(n-1)} ...
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1answer
487 views

Deriving a recurrence relation

The number of sequences of length $n$ consisting of positive integers such that the opening and ending elements are $1$ or $2$ and the absolute difference between any $2$ consecutive elements is $0$ ...
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1answer
121 views

Geometric interpretation of the fundamental theorem for coalgebras?

Given an element $m$ in a coalgebra $C$, there always exists a finite-dimensional subcoalgebra $D \subset C$ containing $m$; this is the fundamental theorem for coalgebras. This obviously isn't the ...
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1answer
166 views

general technique to convert recurrence relation to integral

I know the following recurrence relation $$a_n=\frac{a+na_{n-1}}{a-n}$$ with $a_0=1$ can be represented alternatively as an integral $$a_n=a\int_0^1{x^{a-n-1}(2-x)^ndx}$$ Verifying this is easy, ...
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1answer
253 views

A nicer recurrence for the Eulerian polynomials.

I was perusing the subject of Eulerian polynomials. I'm assuming the definition that the Eulerian polynomial is defined by $C_n(t)=\sum_{\pi\in S_n}t^{1+d(\pi)}$, where $d(\pi)$ is the number of ...
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1answer
213 views

The recurrence $a_k(n) = \sum_{0\leqslant j<n} a_{k-1}(n+j)$

I'm trying to find a closed form expression to the following recurrence relation: \begin{align} &a_k(0) = 0, \quad \forall k\geqslant 1; \\ &a_k(1) = 1, \quad \forall k\geqslant 1; \\ ...
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1answer
200 views

Solving a recurrence with diagonalization?

Considering the recurrence $F_n=F_{n-1}+3F_{n-2}-3F_{n-3}$ where $F_0=0$, $F_1=1$ and $F_2=2$. Use diagonalization to find a closed form expression for $F_n$. So I first continued the recurrence to ...
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1answer
64 views

Recursive identity for elliptic lattice constants $\sum_{\lambda\in\Lambda\setminus0} \lambda^{-2k}$

I am stuck on Exercise 3 in these notes. To keep this question self-contained: we have $\displaystyle\Lambda=\langle\omega_1,\omega_2\rangle=\omega_1\Bbb Z+\omega_2\Bbb Z\subset\Bbb C,$ ...
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4answers
117 views

Homework | Find the general solution to the recurrence relation

A question I have been stuck on for quite a while is the following Find the general solution to the recurrence relation $$a_n = ba_{n-1} - b^2a_{n-2}$$ Where $b \gt 0$ is a constant. I don't ...
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1answer
110 views

Recurrence for random walk

I have the following recurrence which I get when trying to solve a random walk problem given a positive integer $x$. $p_i = \dfrac{p_{i-1}}{2} + \dfrac{p_{i+2}}{2}$ if $0< i < x$ $p_i = 1$ if ...
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1answer
411 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 ...
5
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3answers
333 views

Find inverse for the closed-form expression of linear recurrence relation

I am trying to find an inverse of the following formula: $$ a_n=\frac{2+\sqrt{6}}{4}(1+\sqrt{6})^n+\frac{2-\sqrt{6}}{4}(1-\sqrt{6})^n $$ This formula is a closed-form expression of a linear ...