1
vote
2answers
82 views

Periodic sequence [duplicate]

$(x_n)_n$ is a sequence defined by the relation: $x_{n+2}=|x_{n+1}-x_{n-1}|$ for $n\geq1$ and $x_0,x_1,x_2$ are non-negative integers, not all three equal 0. I think this sequence is periodic, so here ...
0
votes
3answers
100 views

Why is a sequence $(x_1, x_2, x_3,…)$ eventually periodic if the set $\{x_1, x_2, x_3,…\}$ is finite?

"A sequence $(x_1, x_2, x_3...)$ is eventually periodic if the set ${x_1, x_2, x_3,...}$ is finite. (Think about it.)" (Vivaldi) First of all I am not sure whether this is an if and only if statement ...
5
votes
2answers
160 views

If $f$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero.

Please show that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous periodic function with irrational period and if $\sum_n\frac{|f(n)|}{n}<\infty$, then $f$ is identically zero. (For example, using ...
1
vote
0answers
76 views

Is there a generic approach to Generating Function of periodic sequences?

Recently I read on wiki (see here): "Any periodic sequence can be constructed by element-wise addition, subtraction, multiplication and division of periodic sequences consisting of zeros and ones." ...
0
votes
0answers
351 views

Find the period of sequence.

A sequence is such that its terms are generated by the formula: $$r_i =(ar_{i-1}+b\pmod m)$$ where $a,b,m,r_0$ are given. find the period,that is the number of terms that are repeated. For example, ...
0
votes
2answers
392 views

Period of a finite binary sequence

Let $G:N\to\{0,1\}$, and let $L$ be some period of $G$, so that $G(i+kL)=G(i)$. What's the best a good way to find the smallest period of $G$? I mean an algorithm that takes ($G$,$L$) and outputs the ...
1
vote
0answers
105 views

sum of periodic function which eventually vanish

It is known that : If $\lim_{x \to \infty} \sum_{i=1}^n f_i(x)=0$ and $f_i$ are real-valued periodic function, then $\sum_{i=1}^n f_i(x)=0$ for all $x \in \mathbb{R}$. We can solve this problem by ...