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Consider if we have number sequence e.g. $\left\{ 1, 6, 62, 344, ... \right\}$ I'd like to ask if is it possible to compute or estimate form of $n$-th element of this series.

Is it possible with Mathematica or maybe some math processes? I'll be glad for all hints and solution to this topic. Greetings,

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WolframAlpha can often guess, but it doesn't know anything about this. OEIS doesn't find it either – Cocopuffs Apr 12 '13 at 9:32
Where does the sequence come from? Any hint on how it is generated? With that, it might be possible to come up with a formula... – vonbrand Apr 12 '13 at 9:53
@vonbrand This example is "random". I need to design simulator that will count something and the output might be looking like this. I think we can consider that for all $n$ : $a_{n+1} > a_{n} $ and increasing speed will be faster than linear. – piotrdab Apr 12 '13 at 10:13
@Cocopuffs: I don't think WolframAlpha can generate series for $[\sqrt{n}]^2+n$? – Inceptio Apr 12 '13 at 10:21
@Inceptio It looks for "holonomic sequences", which has something to do with the generating power series satisfying a simple (polynomials) differential equation. It doesn't do everything, but it often helps. – Cocopuffs Apr 12 '13 at 10:48

It really depends on how your sequence is generated.

If it comes from a real world problem, the ideal "function" which perfectly models your data is usually too complicated and takes too many parameters into account to even think about inferring it.

However there are a lot of statistical tools at your disposal to try and fit the data to predict further values. The problem you're facing is called "Regression" which is basically trying to find a simpler version of the "ideal function" which will model your data as much as it cans while still optimizing the prediction of future values.

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