## C - Tricolor Pyramid Editorial by evima

Let us call the colors \(0, 1, 2\). Then, on top of two blocks of colors \(a\) and \(b\), a block of color \(-(p_1 + p_2) \bmod 3\) will be placed.

Thus, the answer is \(\left(-1\right)^{N-1} \times \sum_{i=0}^{N} (c_{i-1} \times {}_{N-1}C_{i-1})\), modulo \(3\). (See figures below.)

To compute a binomial coefficient \({}_{n}C_{r}\) modulo \(3\), compute the values \(f(n!), f(r!), f((n - r)!), g(n!), g(r!), g((n - r)!)\), where \(f(m)\) is the number of times \(m\) can be divided by \(3\), and \(g(m)\) is the result, modulo \(3\), of dividing \(m\) by \(3\) until it is no longer divisible by \(3\). For a given integer \(m\), we can compute \(f(m!)\) and \(g(m!)\) in \(O(\log m)\) time.

Sample Implementation in C++: https://atcoder.jp/contests/arc117/submissions/21732692

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