All Questions
Tagged with integer-sequences additive-combinatorics
12 questions
13
votes
2
answers
2k
views
Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
8
votes
0
answers
145
views
Minimum length of sequence such that every integer from 1 to n can be achieved as the sum of some contiguous subsequence
This question literally came to me in a fever dream last night, and it's frustrating me to no end. I'll try to explain it as best I can, but there may not be a satisfying answer; the best outcome ...
7
votes
2
answers
805
views
Distance among integer set
Given an integer set, if the distances between integers in the set are still in the set, what mathematical term should be used to describe that nature? Or what nature does the set have?
For example, $...
- 79
5
votes
1
answer
297
views
Additive basis of order 2
Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ?
Remark : ...
- 655
5
votes
0
answers
307
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 791
4
votes
1
answer
120
views
Avoiding equality of partial sums of two different aperiodic sequences
Consider two distinct sequences of positive integers, $a_{n}|_{n=1}^{\infty}$, and $b_{n}|_{n=1}^{\infty}$ such that for either sequence no period exists. The elements of both sequences are drawn from ...
- 163
4
votes
0
answers
156
views
Inequalities about tripling and doubling sumsets
Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following:
Question 1:...
- 30.6k
3
votes
0
answers
180
views
Additive combinatorics and a Diophantine equation
Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...
- 2,227
2
votes
1
answer
205
views
Difference sequences of sets of integers
In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...
- 995
1
vote
1
answer
115
views
Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
- 3,625
0
votes
1
answer
220
views
Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?
The infinite series representation
\begin{equation}
\frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0
\end{equation}
for the positive ...
- 1,101
0
votes
0
answers
88
views
Infinite difference length of integer subsets
Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...
- 995