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Results tagged with sequences-and-series
Search options questions only
not deleted
user 61426
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
0
votes
1
answer
379
views
A possible surprise involving Euler's constant $e$ [closed]
Let
\begin{align*}
c_n &= n!\left(e-\sum_{k=0}^n \frac{1}{k!}\right) \\
\\
u_n &= \bigg\lfloor{\frac{1}{c_n} \bigg\rfloor} \\
\\
v_n &= \bigg\lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} \bigg\rfloo …
- 3,443
7
votes
1
answer
170
views
Number of numbers in $n$th difference sequence
Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that
…
- 3,443
8
votes
4
answers
343
views
Simple-looking sequences $A$ and $B$ defined by a complementary equation
Define $A=(a_n)$ and $B=(b_n)$ by $b_0=1$ and
$$a_n=b_n+b_{2n}$$
for $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$. Can someone prove t …
- 3,443
12
votes
1
answer
739
views
Possible limit involving the gamma function
Does $$\lim_{n \to \infty} \int_{0}^{1} \Gamma(x)^{n/(n+1)}dx - n$$ exist?
Here's some background. The integral
$$\int_{0}^{1} \Gamma(x) dx$$
diverges rather slowly. Inserting the exponent $n/(n+1)$ …
- 3,443
4
votes
1
answer
203
views
Difference of two integer sequences: all zeros and ones?
Suppose that $c$ is a nonnegative integer and $A_c = (a_n)$ and $B_c = (b_n)$ are strictly increasing complementary sequences satisfying
$$a_n = b_{2n} + b_{4n} + c,$$
where $b_0 = 1.$ Can someone …
- 3,443
4
votes
2
answers
287
views
Positions in the Wythoff array
Suppose that $x$ and $y$ are positive integers. How can the position of $x+y$ in the Wythoff array (A035513) be predicted from the positions of $x$ and $y$?
Background. The Wythoff array begins with …
- 3,443
3
votes
1
answer
312
views
Another question about the golden ratio and other numbers
This is an extension of "A question about the golden ratio and other numbers." Given $r$, suppose that $$c_0+c_1x+c_2x^2+ \cdots = \frac{1} {\lfloor{r}\rfloor+\lfloor{2r}\rfloor x+\lfloor{3r}\rfloor …
- 3,443
20
votes
2
answers
1k
views
A possibly surprising appearance of $\sqrt{2}.$
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs …
- 3,443
13
votes
2
answers
963
views
What is the Hausdorff dimension of this fractal?
Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= \sum_{i=h}^\infty …
- 3,443
5
votes
1
answer
680
views
When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the se …
- 3,443
7
votes
2
answers
428
views
Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs exac …
- 3,443
5
votes
1
answer
302
views
Simply generated sequences with mysterious differences
Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$ and …
- 3,443
2
votes
1
answer
740
views
Power tower made of $2$s and $3$s: too high, too soon?
A power tower of a number $x$ is typified by
$$ 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