2
votes
0answers
161 views

Arithmetical properties of certain recurrence relations

Consider the following recurrence relation: $$(4i+6j+1)(2i+3j)a_{i,j}=3j a_{i+1,j-1}-2i a_{i-1,j}+32i(i-1) a_{i-2,j+1}, i\ge0, j\ge0,$$ $$a_{0,0}=1.$$ This equation appeared in the article ...
8
votes
1answer
441 views

Hamming weight of Fibonacci numbers

The Hamming weight $w(n)$ is the number of 1s in $n$ when written in binary. Is there some effective bound on Fibonacci numbers $F_n$ with $w(F_n)\le x$ for a given $x$? Clearly only $F_0=0$ has ...
1
vote
1answer
176 views

Infinitely many sufficiently large powers in linear recurrences

Edit Aaron solved the original question with the fourth order $$ a(n)=n2^n+\frac{(-1)^n-1^n}{2} $$ trying to make the question harder. Let $a(n)$ be a linear recurrence with constant coefficients, of ...
5
votes
1answer
904 views

Beyond Collatz: A $5n+1$ conjecture? [closed]

Let $$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
6
votes
3answers
542 views

References on techniques for solving equations with discontinuous functions such as floor and ceiling?

Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good ...
6
votes
5answers
459 views

Uniqueness of values in recurrence relations

Given an integer $k > 1$, define the sequences $X(k,n), Y(k,n)$ as follows: $a=4k-2,$ $y_0 = 1,$ $y_1 = a + 1,y_n = ay_{n-1} - y_{n-2}$ $b = 4k + 2,$ $ x_0 = 1,$ $x_1 = b - 1,$ $x_n = ...