1. Let us consider 1 2 3
It is a simple arithmetic progression having common difference, d=1;

2. Let us consider 1 2 4
It is a simple geometric progression having a ratio, r =2;

3. Let us consider 1 2 8

What is it? Would you make it a series of some mathematical function.

I hope you will find a mathematical function, (2)^[{(2)^n}-1] where n stands for 0, 1 and 2
when n=0, then (2)^[{(2)^0}-1] = (2)^[1-1] =2^0=1
when n=1, then (2)^[{(2)^1}-1] = (2)^[2-1] =2^1=2
when n=2, then (2)^[{(2)^2}-1] = (2)^[4-1] =2^3=8.
Let us have fun!!!