All Questions
9 questions
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 ...
- 3,443
35
votes
0
answers
1k
views
Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?
In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.
QUESTION Is there any (added: ...
- 23.7k
4
votes
2
answers
594
views
Squares in Lucas sequences
Good night, everyone!
According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
- 87
0
votes
1
answer
101
views
Recurrence for the number of steps required to get one ball in each box
Given $n$ balls, all of which are initially in the first of $n$ numbered boxes, $a(n)$ is the number of steps required to get one ball in each box when a step consists of moving to the next box every ...
- 4,948
26
votes
1
answer
7k
views
Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
- 4,948
9
votes
2
answers
1k
views
The p-adic valuation of a linear recurrence
Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely,
$$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$
for some $a_1, \ldots, a_k \in \...
user40023
6
votes
1
answer
268
views
Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$
Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$.
The sequence begins with
$$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
- 4,948
3
votes
1
answer
140
views
Sequences that sum up to Dowling numbers
Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
- 4,948
1
vote
2
answers
534
views
Can these sequences stay integer-valued as many times as we want and then fail?
Edit:
Suppose that we choose some integer $d$ and some natural number $c=c_2$. Then if we plug those values into
$$ c_{n+1}=\frac{c_n(c_n+n+d)}n $$ and observe the behavior of this recursively ...
user114642