As already discussed a little bit in the chatroom the following question came to my mind while studying some sequences of natural numbers.
If you need to find closed formulas (not piecewise function or polynomials which interpolate the numbers) for sequences like
the first thing you'll probably try is to find a rule how the numbers are build up. In (i) it seems to be that the next number is the previous number multiplied by the factor 2, in (ii) we always multiply the previous number by 8.
My question is; What is the minimum number of numbers you need to find an unique closed formula expression? Lets take for simplicity only the the natural numbers.
Is there for exmaple a sequence $a(n)=?$ where $a(1)=2,a(2)=4, a(3)=8,a(4)=16$ but $a(5)\not=32$ and if yes is there also a sequence where $a(5)=32$ but $a(6)\not=64$ and so on?