Question 31
Peter gets bored during the lockdown, so he decides to write numbers the whole day. He makes a sequence of numbers starting with $0$, $1$ and $-1$, and then going on indefinitely. On the next line he writes the same sequence of numbers, but shifted one place to the right. On the third line he writes again the same sequence of numbers, shifted another place to the right. He adds all three numbers standing in a vertical column. (He skips the first two places so he starts with $-1 + 1 + 0$.) The answer for every column is the next multiple of three. Peter's paper hence looks like this:
$$
\begin{array}{c|rrrrrrrr}
&0&1&-1&\cdots&\cdots&\cdots&\cdots&\cdots\\
& &0&1&-1&\cdots&\cdots&\cdots&\cdots\\
+& & &0&1&-1&\cdots&\cdots&\cdots\\\hline
& & &0&3&6&9&12&\cdots
\end{array}
$$
The first number in the uppermost sequence is $0$, the second number is $1$, the third number is $-1$, etcetera. Determine the $2021$st number in the uppermost sequence.
$$
\begin{array}{c|rrrrrrrr}
&0&1&-1&\cdots&\cdots&\cdots&\cdots&\cdots\\
& &0&1&-1&\cdots&\cdots&\cdots&\cdots\\
+& & &0&1&-1&\cdots&\cdots&\cdots\\\hline
& & &0&3&6&9&12&\cdots
\end{array}
$$
The first number in the uppermost sequence is $0$, the second number is $1$, the third number is $-1$, etcetera. Determine the $2021$st number in the uppermost sequence.