# Tagged Questions

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

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### A recurrence for a combinatorial problem

$N$ balls are tossed into $n$ boxes independently. Each ball has a $1/n$ chance of falling into any box.$$P_{N,n}(k):= Pr\{exactly\:k\:empty\:boxes\:after\:N\:balls\:thrown\:into\:n\:boxes\}$$ Show ...
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### How to solve the difference equation $u_n = u_{n-1} + u_{n-2}+1$

Given that: $$$$u_n=\begin{cases} 1, & \text{if 0\leq n\leq1}\\ u_{n-1} + u_{n-2}+1, & \text{if n>1} \end{cases}$$$$ How do you solve this ...
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### Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk: A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
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### Wolfram alpha giving wrong result on recurrence?

I have the recurrence$$a_{n+2}-a_n=1$$ The answer I got was $a_n=A+B(-1)^n+\frac n 2$, while WolframAlpha is giving me $a_n=A+B(-1)^n+\frac n 2- \frac 1 4$. Although when I plug them in the relation,...
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### Lines and planes-recursive formula

A family of $n$ lines is drawn in the plane such that each pair of lines cross and no $3$ dinstinct lines have a point in common Let $r(n)$ denote the number of regions into which ...
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### How I can make a proof to this conjecture if it is possible?

Is there someone who can show me the way helping me to proof this conjecture. If it's not open, at a least show me links or papers which to be helpful. Conjecture: Assume $c > 0$ and that an ...
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### Explanation of Linear Nonhomogeneous Recurrence Relations Problem

$$5\times 3^n=v_{n+2}-6v_{n+1}+9v_n$$ $$=C(n+2)^23^{n+2}-6C(n+1)^23^{n+1}+9Cn^23^n$$ $$=18C3^n$$ Can anyone explain to me how he got $18C3^n$. I've been simplifying the 2nd step but haven't gotten ...
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### Looking for a “nice” Recurrence relation…

I'm want to build a game (with steps) that the solution have a Recurrence relation, i.e. - to solve the game you have to move from point A to point B, from point B to point C...(kind of a maze). Of ...
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### Recurrence equation $f(g\cdot x)/f(x)=t$

I am trying to solve equation $\frac{f(g x)}{f(x)}=t$ and I'm out of ideas. Any suggestions? In my problem, $f$ is a continuous and weakly increasing function.
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### Step by step Linear Reccurence

Can someone explain to me in a little bit more detail how you can get to this point. I know its explained here but i'm trying to apply the way he did this problem to this one \begin{equation*} u_n-...
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### How to find the recurrence relation from a given polynomial?

Consider the formal power series: $A(x)=\sum a_nx^n$. and $A(x)= \frac{8+14x-50x^2}{1-7x^2+6x^3}$ I am trying to derive a recurrence relation, Is there a general method for doing it? Please help, ...
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### Proof the golden ratio with the limit of Fibonacci sequence [duplicate]

Let $F_n=F_{n-1}+F_{n-2}$ the Fibonacci numbers, and $\phi=\frac{1+\sqrt5}{2}$ The exercise asks me to prove that: $\lim\limits_{n \to \infty}\frac{F_{n+1}}{F_n}=\phi$. Sorry as can be proceed??