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sequences, u_(2n)=u_n and u_(2n+1)=1=u_n

Source: France 1990 P1

May 18, 2021
algebranumber theorySequences

Problem Statement

Let the sequence unu_n be defined by u0=0u_0=0 and u2n=unu_{2n}=u_n, u2n+1=1−unu_{2n+1}=1-u_n for each n∈N0n\in\mathbb N_0. (a) Calculate u1990u_{1990}. (b) Find the number of indices n≤1990n\le1990 for which un=0u_n=0. (c) Let pp be a natural number and N=(2p−1)2N=(2^p-1)^2. Find uNu_N.
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