How to identify the pattern of a sequence 
Are there some particular methods for identifying the following types of number series?



*

*$6, 10, 19, 44, 93, \cdots$ (Difference being prime no's square starting from 2)

*$1, -2, 15, 52, -512, \cdots $ ( $^*2-4,\ ^*-6+3,\ ^*4-8,\ ^*-10+5$, and so on)

*$4, -2, -7, 25, 95,\cdots$ ( $^*-1+2,\ ^*2-3,\ ^*-3+4,\ ^*4-5$, and so on)

I mean they do not follow the arithmetic or geometric series nor do their common difference seem to follow any AM-GM pattern. So, is there any generalized mathematical theorems on these types of number series? Or we have to do it on a trial & error basis using intuition?


You ask 

is there any generalized mathematical theorems on these types of
  number series?

I'll risk an unsatisfactory answer too long for a comment: essentially, "no". 
The sequences school kids work on often come from arithmetic or geometric series, which is probably why you suggest trying them first. But there are no general rules for "these types of number series".
When mathematicians look for patterns  they usually have some reason to expect a particular form, so their intuition informs the search.  Knowing the source of a sequence in advance really matters. If I encountered your first one while thinking about number theory I might guess something involving primes after I noticed that the differences were all squares but that 16 and 36 were missing.
Any finite sequence can be continued in many ways that look as if they extend a pattern - you can always do this with a polynomial by taking enough differences. (This google search for successive differences polynomial finds lots of links.)
When a mathematician thinks she's found a new pattern she then tries to prove it goes on forever - that requires more than checking the next few terms. It's fun.
