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strange sequence with floor function

Source: Oliforum Contest I 2008 2.2 https://artofproblemsolving.com/community/c2487525_oliforum_contes

September 28, 2021
floor functionalgebrafunction

Problem Statement

Let {an}n∈N0 \{a_n\}_{n \in \mathbb{N}_0} be a sequence defined as follows: a1=0 a_1=0, an=a[n2]+(−1)n(n+1)/2 a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}, where [x] [x] denotes the floor function. For every k≥0 k \ge 0, find the number n(k) n(k) of positive integers n n such that 2k≤n<2k+1 2^k \le n < 2^{k+1} and an=0 a_n=0.
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