MathDB

Define the function

g(\cdot): \mathbb{Z} \to \{0,1\}

such that g(n) \equal{} 0 if

n < 0

, and g(n) \equal{} 1 otherwise. Define the function

f(\cdot): \mathbb{Z} \to \mathbb{Z}

such that f(n) \equal{} n \minus{} 1024g(n \minus{} 1024) for all

n \in \mathbb{Z}

. Define also the sequence of integers

\{a_i\}_{i \in \mathbb{N}}

such that a_0 \equal{} 1 e a_{n \plus{} 1} \equal{} 2f(a_n) \plus{} \ell, where \ell \equal{} 0 if \displaystyle \prod_{i \equal{} 0}^n{\left(2f(a_n) \plus{} 1 \minus{} a_i\right)} \equal{} 0, and \ell \equal{} 1 otherwise. How many distinct elements are in the set S: \equal{} \{a_0,a_1,\ldots,a_{2009}\}? (Paolo Leonetti)

Problem Statement