Find the number of positive integers
Source: IMO Shortlist 1994, N4
October 22, 2005
number theoryIMO ShortlistSequencerecurrence relation
Problem Statement
Define the sequences as follows. a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1.
If is even then a_{n \plus{} 1} \equal{} \frac {a_n}{2}, b_{n \plus{} 1} \equal{} 2b_n, c_{n \plus{} 1} \equal{} c_n.
If is odd, then a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n, b_{n \plus{} 1} \equal{} b_n, c_{n \plus{} 1} \equal{} b_n \plus{} c_n.
Find the number of positive integers such that some a_n \equal{} 0.