All Questions
110 questions
6
votes
0
answers
257
views
Convergence with the recurrence $T_{n+1}=T_n^2-T_n+\frac{n}{p_n}$
For each integer $n\geq 1$ I define the recurrence $$T_{n+1}=T_n^2-T_n+\frac{n}{p_n},$$
with $T_1=1$, where $p_k$ denotes the $k$-th prime.
So multiplying by $(-1)^n$ and telescoping gives that for ...
- 63.9k
11
votes
1
answer
418
views
Is Somos-8 $\mod 2$ periodic?
It is known that the Somos-$k$ sequences for $k\ge 8$ do not give integers. But the first terms of Somos-8 sequence $s_n=a_n/b_n$
$$1, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 25, 61, 187, 775, 5827, 14815,\...
- 34.3k
1
vote
0
answers
531
views
Two conjectural identities involving $\zeta(3)$ and the golden ratio $\phi$
Let $\phi$ be the famous golden ratio $\frac{\sqrt5+1}2$, and let
$\zeta(3)=\sum_{n=1}^\infty\frac1{n^3}.$
In 2014, in the paper
Zhi-Wei Sun, New series for some special values of $L$-functions, ...
- 35.5k
2
votes
3
answers
519
views
Does this deceptively simple nonlinear recurrence relation have a closed form solution?
Given the base case $a_0 = 1$, does $a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, ...
- 387
4
votes
1
answer
168
views
An inequality involving $k$-generalized Fibonacci numbers
I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
- 61.9k
2
votes
1
answer
740
views
Power tower made of $2$s and $3$s: too high, too soon?
A power tower of a number $x$ is typified by
$$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$
Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers
$$...
- 3,443
5
votes
1
answer
303
views
Simply generated sequences with mysterious differences
Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$...
- 60.5k
7
votes
2
answers
428
views
Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
- 60.5k
18
votes
1
answer
626
views
For a linear recurrence sequence $(u_n)_{n\geq 0}$, can $\{i \mid u_i > 0\}$ be the set of Fibonacci numbers?
Question: Is there a linear recurrence sequence $(u_n)_{n\geq0}$ (on the rationals, but I would also be interested by reals) for which $\text{Pos}(u) = \{i \mid u_i > 0\}$ is precisely the set of ...
CommunityBot
- 1
7
votes
1
answer
4k
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 ...
CommunityBot
- 1