Well, I have the following two problems involving Fibonacci sequences and Lucas numbers. I know that they share the same technique, but I don’t have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 =0, f_1=1$$ $$l_n=l_{n-1} +l_{n-2}:l_0=2,l_1=1$$ Now, I want to prove that: $$\sum\limits_{k=0}^nf_k= f_{n+2}-1 $$ $$\sum\limits_{k=0}^n l_k^2= l_nl_{n+1} +2$$ My question is, what […]

I am working on the following problem on 2-coloured complete graphs: $K_9$ is coloured red and blue and contains no red triangle and no blue $K_4$ then every vertex must have red degree 3 and blue degree 5. Is this possible? I am pretty sure the answer is ‘no’ but I am not sure how […]

This question already has an answer here: Combinatorial proof of binomial coefficient summation 2 answers

I’m trying to find the smallest graph that is regular but not vertex-transitive, where by smallest I mean “least number of vertices”, and if two graphs have the same number of vertices, then the smaller is the one with the lower number of edges. I currently have that the smallest such graph is the disjoint […]

My problem is similar to this one, but different in some significant ways. As in the above question, I have voting with $n$ voters and $m$ candidates. However, I care about which voter voted for which candidate. As such, there are a total of $m^n$ possible configurations. Also, of these possible configurations, how many did […]

Let $\Sigma$ be an infinite set. Let $A,B \subseteq \Sigma$ be of finite symmetric difference iff they have a finite difference, more formally: $A \sim B$ iff $|A \Delta B| \in \mathbb{N}$ How many equivalence classes does $\sim$ have?

While doing some Computer Science problems, I found one which I thought could be solvable using combinatorics instead of programming: Given two positive integers $n$ and $k$, in how many ways do $k$ numbers, all of which are between $0$ and $n$ (inclusive), add up to $n$? In the problem, order does matter. For example, […]

How to prove $$\sum_{k=0}^{r}\binom{r}{k}(r-k-1)^{r-k}(k-1)^{k-1}+(r-2)^r=0$$ I met this function when I tried to give another proof of the known lower bound of Tur\’an functions of complete hypergraphs ( Based on a same construction, instead of using shifting method, I tried to count edges directly ) Here is the question: Define $a_0=-1$ and $a_1=1$. For all $r\geqslant2$, […]

32 people were invited at a party and started exchanging handshakes. Because of the confusion, each of them shook hands with each other multiple times: at least twice and up to X times. However, every two people exchanged different number of handshakes from every other two. What is the minimum possible number X, so that […]

So this question has probably been answered already, but I can’t find a good answer through searching google or this site. Basically, if I have n symbols, how many n-length combinations of the symbols can I make, excluding symmetrical duplicates and duplicates made by switching symbols for each other? For instance, with the following sets […]

Intereting Posts

Exactly one nontrivial proper subgroup
Find an equation of the plane passing through 2 points and perpendicular to another plane
Equivalent Norms on Sobolev Spaces
Does Lowenheim-Skolem theorem depend on axiom of choice?
How to solve this equation for $x$ with XOR involved?
Clarification on the meaning of dx in the integral and differential setting
Probability Distribution of Rolling Multiple Dice
How many triangles with integral side lengths are possible, provided their perimeter is $36$ units?
Indefinite integral of product of CDF and PDF of standard normal distribution
existence of the solution of Neumann problem in $\mathbb{R}^3$
Rationalizing expressions
Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
How to show that $f$ is an odd function?
Prove that bitstrings with 1/0-ratio different from 50/50 are compressable
What Does it Mean for a Function to have Finite Support?