Binomial Coefficients

For a positive integer n, let \(C_n=\dfrac{1}{n+1} \binom{2n}{n}\) , and \(S_n=C_1 + C_2 + \ldots + C_n \) .

Let's say \(S_{n} \equiv 1 \pmod{3} \) if and only if there exists a "certain digit" in the base 3 expansion of \(n + 1\).

Find that "certain digit".

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