2
votes
2answers
74 views

Solution of an equation involving even integers

If $x$ is any positive even integer $> 1$. I compute: $$ c = x + x! $$ Now assume instead $c$ (even integer) is given, and I want to get back the value $x$. Is it possible to find a simple ...
3
votes
2answers
43 views

A sum of difference of floors

I have the sum ( $M$ is any integer $> 1$ ): $$ \sum_{h = 1}^{M}\left(\,\left\lfloor\, 2M + 1 \over h\,\right\rfloor -\left\lfloor\, 2M \over h\,\right\rfloor\,\right) $$ and looking for a way to ...
2
votes
2answers
71 views

Sum of floor of ratios

I need to compute, in a program at work, the sum, for $k = 2$ to $n-1$, of the floors of the ratios: $\frac{n}{k}$. Since n is a large integer in my case I would need a "closed form" formula for this ...
5
votes
1answer
70 views

Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2) $$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
1
vote
2answers
54 views

Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
4
votes
1answer
73 views

Closed form of $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$

Recently, I came across the following exercise on the course of discrete math Find a closed form for $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$ So I tried some of the usual techniques: Let ...
4
votes
5answers
323 views

Find a closed expression for a formula including summation

Let: $$\sum\limits_{k = 0}^n {k\left( {\matrix{ n \cr k \cr } } \right)} \cdot {4^{k - 1}} \cdot {3^{n - k}}$$ Find a closed formula (without summation). I think I should define this as a ...
-3
votes
1answer
48 views

Closed form questions [closed]

Please could you help me to find the generating functions of the following sequences in closed form: (a) 1, 0, 1, 0, 1, 0, … (b) 2, –4, 6, –8, 10, –12, …
0
votes
1answer
45 views

The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

Is there a closed form for $\sum_{n=0}^{C} mod(n,m)$ for arbitrary integers m ?
1
vote
4answers
110 views

How to solve a recursive equation

I have been given a task to solve the following recursive equation \begin{align*} a_1&=-2\\ a_2&= 12\\ a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2, \quad n \geq 3. \end{align*} Should I start by ...
2
votes
4answers
586 views

How to find a closed form solution to a recurrence of the following form?

I need to find the closed form solution to the following recurrence -: $ T(n) = 8*T(n/2) + 0.25*n^2$ with $T(1) = 1$ and $n=2^j$ and this is what I have tried so far but just can't seem to get a ...
1
vote
1answer
105 views

Recursive and closed form solution for choosing $n$ pairs/triplets.. of $kn$ elements.

I stumbled apon an interesting question: How many ways are there to arrenge $kn$ elements into $n$ sets, $k$ elements each? There should be a recursive and closed form solution for $g_k(n)$. For ...
2
votes
1answer
251 views

Recurrence relation for $n$ numbers in which no 3 consecutive digits are the same.

I am stuck on trying to find (and solve) a recurrence relation to find all n-digit numbers in which no 3 consecutive digits are the same. These numbers are in decimal expansion. Now I first ...
3
votes
1answer
246 views

Finding a Linear Recurrence Relation

A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. ...
0
votes
4answers
361 views

Having a lot of trouble solving this recurrence with iteration and finding a closed form…

I'm learning discrete math and didn't have any trouble with any recurrences in the examples I went over through the chapters on it, but this one problem at the end of the first chapter is killing me, ...
2
votes
0answers
148 views

Can this series be expressed in closed form, and if so, what is it?

Can this series be expressed in closed form, and if so, what is it? $$ \sum_{n=1}^\infty\frac{1}{9^{n+1}-1} $$
2
votes
2answers
296 views

Closed-form Expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$? (problem with Mathematica)

I need to calculate a closed-form expression for $\sum_{j=0}^{k-1}(2j+2)\sum_{i=1}^j \frac 1 {i^2}$. This isn't particularly difficult, and I do it by hand pretty much routinely. However I found out ...