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Prove the formula for A(n)

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September 1, 2010
trigonometryalgebraSequencecountingrecurrence relationIMO ShortlistIMO Longlist

Problem Statement

Denote by ana_n the greatest number that is not divisible by 33 and that divides nn. Consider the sequence s0=0,sn=a1+a2+⋯+an,n∈Ns_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N. Denote by A(n)A(n) the number of all sums sk (0≤k≤3n,k∈N0)s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0) that are divisible by 33. Prove the formula A(n)=3n−1+2⋅3(n/2)−1cos⁡(nπ6),n∈N0.A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.
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