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.
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 $$y_k={1\over3}k^2 + k + {7\over9} + {2\over9} \cos{2k\pi\over3}\qquad(0\leq k\leq9)\ .$$