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Is it possible to figure out what function produces these values?

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Repeating intervals like that are a characteristic of the floor function, so I'd guess something like

x - 7[x/7] should give a similar graph. You could then split the function at x = -1 and represent the point symmetry by f(x) -> -f(-x).

Though for a side note, this isn't finding the "function" that describes the values, the chart (and pattern) you plotted itself is the function. You could describe that function in any way you'd like, this would just be coming up with an elegant way to write it.

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