There´s a very common finite series, that use to be, at the begining on every book:

Where can be real or complex.

There is a, very well known, particular case of this series where :

are the Mersenne Numbers, and due to this sum, is easy to see that the **Mersenne numbers** consist of all 1s in base-2 (they are base 2 repunits)

But this entry is about another particular case of this finite sum:

Where: , is the complex unit.

This sum shows periodical behaviour with a period of , and its values changes from one vertex to another in a square of side equal to , if we plot them in the complex plane:

if

if

if

if

If we take a look at the real part of the complex number :

Then we had just found the sequence A133872 from OEIS, and then we can construct another expressions for this sequence, and also for the problem series:

Then, if we expand to trigonometrical functions :

And finally using the information inside OEIS:

**References:**[1]-A133872-Period 4: repeat 1,1,0,0. The On-Line Encyclopedia of Integer Sequences!

### Like this:

Like Loading...

*Related*