= Beatty sequence reciprocity
{c}
= Beatty reciprocity
{c}
{synonym}
Let $\alpha>1$ be irrational and $\beta=\alpha/(\alpha-1)$, so $1/\alpha+1/\beta=1$. The <Beatty sequences> $\lfloor n\alpha\rfloor$ and $\lfloor n\beta\rfloor$, $n\geq1$, partition the <positive integers>.
Indeed, the numbers of terms at most $m$ are $\lfloor(m+1)/\alpha\rfloor$ and $\lfloor(m+1)/\beta\rfloor$. Their arguments sum to the integer $m+1$, and neither is <integral>, so the counts sum to $m$. Subtracting the count at $m-1$ shows that every integer occurs exactly once in the two lists together. This proof interprets reciprocity as an exact complementarity of two densities.
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