Dear HK,

I might not have made myself quite clear. The thing is that exactly the same procedure as in my post works with your sequence.

data = Flatten[Import["~/Desktop/sequences/sequence-1.txt", "CSV"]];
interpol = InterpolatingPolynomial[data, t];

Also works for your sequence of 1500 or so numbers.

Table[Abs[Floor[interpol]], {t, 1, 1696}]

gives exactly the right sequence! The point that you only want positive numbers is irrelevant, because it is easy to make a rule - like I did above - where you take the Abs of everything.

Of course, this will eventually go to infinity, but even that can be cured very easily; you could for example use some Fourier type thing - or simply say that the sequence is constant after some transient. The thing is that this does not seem to be the point. We can always find an infinite number of ways of writing this.

There are some other interesting facts here:

1) If you look at the difference of consecutive data points

ListPlot[Differences[data]]

this looks very much exponentially distributed.

Histogram[Differences[data]]

2) Run

EstimatedDistribution[Differences[data], ExponentialDistribution[[Mu]]]

and you get

ExponentialDistribution[0.00169902]

This also shows that your time series is monotonously increasing.

3) Run a hypothesis test to see whether the Exponential Distribution works:

DistributionFitTest[Differences[data], ExponentialDistribution[0.0016990249902017881`], "HypothesisTestData"]["TestDataTable"]

or alternatively

DistributionFitTest[Differences[data], ExponentialDistribution[0.0016990249902017881`],  "HypothesisTestData"]["TestConclusion"]

gives

4) The autocorrelation function also drops like a stone:

ListPlot[Transpose[{Range[21] - 1, CorrelationFunction[Differences[data], {0, 20}]}] // N, PlotRange -> All]

so consecutive values are practically uncorrelated.

Ok, then. What can we conclude? This looks very much like a stochastic process; similar to the stuff you would see in radioactive decay:

http://community.wolfram.com/groups/-/m/t/250923

You also find similar data when you look at arrival times of customers, or telephone queues etc.

One can find - that is actually always the case - an infinite number of formulas that describe that sequence, but from the data that appears to be futile, because that description will need more or less as many parameters as you have numbers in the sequence.

What would help is to know where that data comes from. Is it some kind of measurement of something? If so, of what?

Cheers, Marco

Are you sure such formula exists? Even The On-Line Encyclopedia of Integer Sequences does not have it or its simplified version:

# - #[[1]] &@{105208, 105508, 105637, 105934, 106208, 106258, 106377, 106769, 107293, 107537}

{0, 300, 429, 726, 1000, 1050, 1169, 1561, 2085, 2329}

Explaining what is the nature of the numbers, giving more numbers, and any additional information - desirably complete info about the problem - would be helpful.