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sequence & representation

Source: germany 2002

August 16, 2005

algebra unsolvedalgebra

Problem Statement

We consider the sequences strictely increasing

(a_0,a_1,...)

of naturals which have the following property : For every natural

n

, there is exactly one representation of

n

a_i+2a_j+4a_k

, where

i,j,k

can be equal. Prove that there is exactly a such sequence and find

a_{2002}