Suppose I have an "arbitrary" sequence of numbers and as a convenience, I want some type of concise formula to be able to generate/regenerate that exact sequence. How can I do it? Can this be done for any sequence or are there limitations? In math, can this "sequence generator" have a bunch of if statements to handle "sub-sequences" and/or erratic shifts?
Here is a sequence so how would I find a formula for this and in general, other similar sequences?
$200, 100, 50, 60, 75, 39, 15, 35$
On
To simplify it some we can "cheat" and do something like this:
A[1] = 200
A[2..3] = 1/2 A[x-1]
A[4..8] = -(31 x^4)/24+(281 x^3)/12-(3209 x^2)/24+(3259 x)/12-100
but plugging in 1 for sequence 4 (value 60), 2 for sequence 5 (value 75)...
$$-\frac{31 x^4}{24} + \frac{281 x^3}{12} - \frac{3209 x^2}{24} + \frac{3259 x}{12} - 100 $$
or (since the denominators are very similar to each other)
$$\frac{-31 x^4 + 562 x^3 -3209 x^2 + 6518 x}{24} - 100 $$
For any finite sequence of real numbers (as in your example), there is a formula that exactly reproduces it, e.g. an interpolating polynomial.
For your example sequence $[200,100,50,60,75,39,15,35]$ (indexed from $1$ to $8$), the lowest degree interpolating polynomial is
$$-\frac{163 x^7}{2520} + \frac{17 x^6}{9} - \frac{3947 x^5}{180} + \frac{9337 x^4}{72} - \frac{151531 x^3}{360} + \frac{55867 x^2}{72} - \frac{360187 x}{420} + 593 $$
which exactly reproduces the given sequence for $x=1,2,3,4,5,6,7,8$.