MathDB
TOT 1999 Spring AS5 sequence with parity of 1s in binary representation

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May 11, 2020
number theorydecimal representationBinarypower of 2Sequence

Problem Statement

For every non-negative integer ii, define the number M(i)M(i) as follows: write ii down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set M(i)=0M(i) = 0, otherwise set M(i)=1M(i) = 1. (The first terms of the sequence M(i)M(i), i=0,1,2,...i = 0, 1, 2, ... are 0,1,1,0,1,0,0,1,...0, 1, 1, 0, 1, 0, 0, 1,... ) (a) Consider the finite sequence M(O),M(1),...,M(1000)M(O), M(1), . . . , M(1000) . Prove that there are at least 320320 terms in this sequence which are equal to their neighbour on the right : M(i)=M(i+1)M(i) = M(i + 1 ) . (b) Consider the finite sequence M(O),M(1),...,M(1000000)M(O), M(1), . . . , M(1000000) . Prove that the number of terms M(i)M(i) such that M(i)=M(i+7)M(i) = M(i +7) is at least 450000450000.
(A Kanel)
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