Looking for insights about this sequence:

In summary, the terms in this series are defined by the equation f(n)*f(n+1)=n, where n is an integer. The top row represents the term number (n), while the bottom row represents the actual value of the term. The closed forms for the even and odd terms are 4^n n!^2 / (2n)! and (2n+1)! / n!^2 / 4^n, respectively.
  • #1
realitybugll
40
0
1 2 3 4 5 6
1, 1, 2, 3/2, 4/(3/2), 5/(4/(3/2)), ...

I suppose there are a lot of variations, but the general idea is the terms are defined by:

f(n)*f(n+1)=n, where n is an integer. The top row is the term # (n), and the bottom one is the actual value of the term

Particularly, I am looking for a way to find the partial sums of the terms.Any responses are appreciated. Sorry for the unclear formatting...
 
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  • #2
realitybugll said:
1 2 3 4 5 6
1, 1, 2, 3/2, 4/(3/2), 5/(4/(3/2)), ...

I suppose there are a lot of variations, but the general idea is the terms are defined by:

f(n)*f(n+1)=n, where n is an integer. The top row is the term # (n), and the bottom one is the actual value of the term

Particularly, I am looking for a way to find the partial sums of the terms.Any responses are appreciated. Sorry for the unclear formatting...

Here's a start; closed forms for the odd and even terms. However, I have started your series counting from 0. That is, where you write f(1) I have f(0). Otherwise it's the same.
[tex]\begin{align*}
f(2n) & = 4^n n!^2 / (2n)! \\
f(2n+1) & = (2n+1)! / n!^2 / 4^n
\end{align*}[/tex]​
Try it.

Cheers -- sylas
 
Last edited:
  • #3
sylas,

Wow! thank you.

Could you maybe give me a hint as to how you found that?
 
  • #4
realitybugll said:
sylas,

Wow! thank you.

Could you maybe give me a hint as to how you found that?

Sure. Basically, I noted that the form of your fractions is as follows.
[tex]\frac{n(n-2)(n-4)(n-6)...}{(n-1)(n-3)(n-5)(n-7)...}[/tex]​
To make life easy for myself, I started by looking only at even values of n. What I actually did was consider cases for 2n, so the equation becomes
[tex]\frac{2n(2n-2)(2n-4)(2n-6)...2}{(2n-1)(2n-3)(2n-5)(2n-7)...1}[/tex]​
This looks a lot like a factorial, so I immediate thought about dividing everything by 2. Once I have a formula for the top line, I can see I will be able to get the bottom line by dividing (2n)! by the top line, so I just focus on the top line. There are n terms being multiplied, so the top line is
[tex]2^n n(n-1)(n-2)(n-3)...1 = 2^n n![/tex]​
The bottom line is therefore
[tex]\frac{(2n)!}{2^n n!}[/tex]​
So we divide these two, and obtain:
[tex]\frac{(2^n n!)^2}{(2n)!} = \frac{4^n n!^2}{(2n)!}[/tex]​
Given the way you numbered equations, this would actually be term number 2n+1. So I simply started numbering from 0. I'm also a pure mathematician, and for programming I like C better than Fortran... so I usually start counting from zero anyway. It often simplifies a problem like this.

Finding the equation for the odd terms was a breeze, using f(n)*f(n-1) = n, which you was your defining relation (adjusted to start counting at zero).

Cheers -- sylas
 
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  • #5
Thanks, I read over it and then went through it myself and got the same thing :approve:
 
Last edited:

1. What is a sequence in science?

A sequence in science refers to a specific order in which events or objects occur. It can also refer to a series of steps or actions that are taken to achieve a certain outcome.

2. How do scientists look for insights in a sequence?

Scientists use various methods such as data analysis, experimentation, and observation to look for insights in a sequence. They may also use mathematical models or computer simulations to better understand the sequence.

3. What is the importance of finding insights in a sequence?

Finding insights in a sequence can help scientists understand patterns, relationships, and underlying mechanisms that may not be immediately apparent. This can lead to new discoveries and advancements in various fields of science.

4. What are some common tools or techniques used to analyze a sequence?

Some common tools and techniques used to analyze a sequence include statistical analysis, bioinformatics, machine learning, and visualization. These tools help scientists identify patterns and trends in the data and make predictions about future events.

5. Can insights from one sequence be applied to other sequences?

Yes, insights from one sequence can often be applied to other sequences, especially if they share similar characteristics or patterns. This can help scientists make broader conclusions and theories about the natural world.

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