All Questions
18
questions
27
votes
2
answers
1k
views
A nice formula for the Thue–Morse sequence
The Thue–Morse sequence$^{[1]}$$\!^{[2]}$ $t_n$ is an infinite binary sequence constructed by starting with $t_0=0$ and successively appending the binary complement of the sequence obtained so far:
$$\...
- 47.3k
4
votes
2
answers
403
views
Maximum run in binary digit expansions
For numbers between $2^{k-1}$ and $2^{k}-1$, how many have a maximum run of $n$ identical digits in base $2$? For instance, $1000110101111001$ in base $2$ has a maximum run of 4.
See picture below ...
- 3,645
2
votes
2
answers
705
views
Upper bound for the strict partition on K summands
In number theory and combinatorics, a partition of a positive integer $n$, also called an integer partition, is a way of writing $n$ as a sum of positive integers. Partitions into distinct parts are ...
- 85
5
votes
1
answer
5k
views
Expressing a positive integer as a sum of positive integers
I am trying to find a way for the positive integers written as the sum of other positive integers.( expressed in terms of some functions)
I searched a bit and I came across with Partitions
But in my ...
- 485
3
votes
3
answers
836
views
How many different ways can a number N be expressed as a sum of K different positive integers?
How many different ways can a number $n \in N$ be expressed as a sum of $k$ different positive numbers?
- 85
2
votes
2
answers
117
views
There exists more than one set of numbers $(m,n)$ so that $132m+34n=\gcd(132,34)$?
Does there exists more than one set of numbers $(m,n)$ so that $132m+34n=\gcd(132,34)$?
I was comparing to, and thinking about how and why euclids algorithm solves for one set.
How might one find ...
- 33
2
votes
1
answer
151
views
The number of ways to represent $(n^2+n)/4$ as a sum of $n/2$ distinct integers in $1,\dots,n$
For any positive integer $n$ (using integer division only), let $P(n)$ denote the number of ways in which the integer $(n^2+n)/4$ can be expressed as a sum of exactly $n /2$ distinct elements of the ...
2
votes
2
answers
588
views
How many ordered quadruples $(w, x, y, z)$ of non-negative integers are there such that $wxyz = 288$? Why?
I found the prime factorization
$$288 = 2^5 \cdot 3^2$$
and then I tried to set up some algebraic equations, but got stuck. How would we proceed so that we get the answer? Especially when we are ...
- 2,042
9
votes
1
answer
899
views
Expected Value for the Number of Parts of a Random Partition (Considering Only a Portion of the Partition Spectrum)
Let $n$ be a positive integer. If we take the set of all partitions of $n$ and choose a random partition from it (uniformly), then the expected value of the number of parts of this partition is a ...
- 113
8
votes
5
answers
312
views
Show that $ \sum_{k=0}^{n} \binom{2n+1}{2k} 3^k $ is divisible by $2^n$
I want to prove that
$$
\sum_{k=0}^{n} \binom{2n+1}{2k} 3^k = \sum_{k=0}^{2n} \binom{2n}{k} 3^{\lceil k/2 \rceil}
$$
is divisible by $2^n$.
I tried induction the following way
\begin{align*}
\...
- 18.2k
6
votes
1
answer
213
views
Periodic sequences resulting from a summation over the Thue–Morse sequence
Let $s_2(n)$ denote the sum of digits of $n$ in base-2 (OEIS sequence A000120), and $t_n=(-1)^{s_2(n)}$. Note that $t_n$ is the signed Thue–Morse sequence (OEIS sequence A106400), satisfying the ...
- 47.3k
5
votes
2
answers
3k
views
Pigeonhole Principle - numbers between $1$ and $100$ [duplicate]
Of the set $A=${$1,2,...,100$}, we will choose $51$ numbers. Prove that, among the $51$ chosen numbers, there are two such that one is multiple of the other
My notes:
1) There are $25$ prime numbers ...
- 3,217
3
votes
3
answers
354
views
why $m$ power by $n$ equals sum of $n$ numbrs
$$m^n=\sum_{i=0}^n(m-1)^i\binom{n}i$$
(a) I want to find a formula for the above and then prove it by induction. But there is two variable right those are $m$ and $n$. I know that this is true, ...
- 1,680
3
votes
2
answers
1k
views
Weak $k$-compositions with each part less than $j$
I am trying to figure out a problem from Richard Stanley's $\textit{Enumerative Combinatorics}$, which has to do with weak compositions of $n$ (sequence of nonnegative integers whose sum adds up to $n$...
- 31
2
votes
1
answer
746
views
Questions on Erdős–Ginzburg–Ziv theorem for primes and understanding related lemmas and their applications.
While trying to prove the prime case of Erdős–Ginzburg–Ziv theorem:
Theorem: For every prime number $p$, in any set of $2p-1$ integers, the sum $p$ of them divisible by $p$.
I came across with ...
- 6,810