Is there a general mathematical method that determines whether any sequence of natural numbers is generated by a particular mathematical law? My question is:

In mathematics is there a general method that determines whether any sequence of natural numbers is generated by a particular mathematical law/function/closed-form expression/recurrence-relation?

What I'm looking for is not how this is a function or closed-form expression. I am simply looking for a general method of determining whether the given sequence is distributed by a certain mathematical law or by random.
Example; 
For sequence $a_n$, 

$$a_n=\left\{ 1,1,2,4,7,11,16,22,29,37,46,56,67,79...\right\}$$

there is a "mathematical relation" : $a_n=a_{n-1}+n-1$

What I want to know is to determine whether this formula exists without having to find any formulas.

Is there such a mathematical/statistical method?
For example, is it possible to do this by visualizing any series?
 A: 
What I want to know is to determine whether this formula exists
  without having to find any formulas.

I pasted those numbers into OEIS and got A152947.
This was a database lookup and I was lucky that the OEIS database
contained an entry to your example.
Further I got only one hit. In general there might be many sequences which share your sample numbers. 
It should be clear that finite samples in general are not specific enough to pin down all possible sequences to a specific one.
E.g. your next element, which you have chosen not to publish, but have in stock, might have been $42$ instead of $92$.
E.g. notice that there are various ways to perform an interpolation to your sample set, e.g. using Lagrange interpolation. Such a function is able to reproduce your given sample set and still might differ from the sequence you had in mind, measured, or have been given in some other way.

I am simply looking for a general method of determining whether the
  given sequence is distributed by a certain mathematical law or by
  random.

Have a look here.
For the case that you expect uniform distribution of a random source, a simple method is using the sequence numbers as coordinates and plot them. If it is uniform you would expect to see no geometric pattern.
This was suggested e.g. by Donald E. Knuth. (Plot on a sphere, Plots of random TCP/IP Initial Sequence Numbers from various operating systems)
