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I got this term¹ as a partial result

$\qquad\displaystyle T = -1 + (-1)^{\frac{1}{3} \cdot (1 - 2 n)} - (-1)^{\frac{2}{3} - \frac{4 n}{3}}$

which Mathematica refuses to (full)simplify further, even assuming that $n$ is a positive integer.

By inspecting a plot and verifying by FullSimplify[T , Mod[n, 3] == k] for k = 0,1,2, I get that

$\qquad\displaystyle T = \begin{cases} -3, &n \mathbin{\mathrm{mod}} 3 = 2 \\ 0, &\text{else} \end{cases}$.

Is there a way to tell (Full)Simplify to prefer case distinctions with simple cases over (complicated) closed forms, or extract a piecewise definition with another function? Preferably without knowing the case conditions a priori, of course.

  1. -1 + (-1)^(1/3 (1 - 2 n)) - (-1)^(2/3 - (4 n)/3)
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