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I have a feeling that it's obvious but I can't find it. I was wondering if anybody else could decode this cryptic thing.

$[1,2,4,7,10,14,19,24,30,37]$

This is very important, I hope someone can see something that I can't.

Also if possible, try to construct an algorithm that can actually predict the one, because I doubt I would have much luck with that either.

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    Oh..my..god!ALEX YOUR A GENIUS! You just solved my problem for the third level, now i gotta boggle my mind over the fourth D:.2012-10-07

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Compute the first and second differences of the given data: $\matrix{ 1&&2&&4&&7&&10&&14&&19&&24&&30&&37 \cr &1&&2&&3&&3&&4&&5&&5&&6&&7&\cr &&1&&1&&0&&1&&1&&0&&1&&1&&\cr}$ Here the third line seems periodic with period $3$ with mean ${2\over3}$. Therefore the given data $(y_k)_{0\leq k\leq9}$ can be produced by a function $k\mapsto f(k)$ of the form $f(k)={1\over3} k^2 + a\ k+ b+ c\ \cos{2k\pi\over3}+ d\ \sin{2k\pi\over3}\ .$ Now fix the undetermined coefficients $a$, $b$, $c$, $d$ such that for $k=0,\ldots,3$ the correct values $y_k$ are produced. The final result is $y_k={1\over3}k^2 + k + {7\over9} + {2\over9} \cos{2k\pi\over3}\qquad(0\leq k\leq9)\ .$

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    @aayush: The $y_k$ satisfy the linear inhomogeneous difference equation $y_{k+2}-2y_{k+1}+y_k=$$a$given periodic function of period $3$. The general theory about such equations says that our "Ansatz" with undetermined $a$, $b$, $c$, $d$ works in this case.2012-10-07
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The On-line Encyclopedia of Integer Sequences has a few results for this sequence.