1992 Putnam A5

For each positive integer $n$, let $a_n = 0$ (or 1) if the

number of 1's in the binary representation of $n$ is even (or odd),

respectively. Show that there do not exist positive integers $k$ and $m$

such that

$$

a_{k+j} = a_{k+m+j} = a_{k+2m+j},

$$

for $0 \leq j \leq m-1$.

对于每个正整数 $n$，令 $a_n = 0$(或 1)，如果

$n$ 的二进制表示中 1 的数量是偶数(或奇数)，

分别。证明不存在正整数$k$和$m$

这样

$$

a_{k+j} = a_{k+m+j} = a_{k+2m+j},

$$

对于 $0 \leq j \leq m-1$。