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As Gerry points out, the sequence $$a_n = [n e] - [(n-1)e],$$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$), and they can be generated quickly from the continued fraction expansion (of $e$, in this case). If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in this article, which was published in Integers. This gives you quickly a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac 1n \sum_{i=1}^n b_i \to e$. I don't have an answer for that. Yet.

As Gerry points out, the sequence $$ a_n = [n e] - [(n-1)e], $$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$), and they can be generated quickly from the continued fraction expansion (of $e$, in this case). If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in my article Fraenkel's Partition and Brown's Decomposition, which was published in Integers (pdf). This gives you quickly a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac 1n \sum_{i=1}^n b_i \to e$. I don't have an answer for that. Yet.

As Gerry points out, the sequence $$a_n = [n e] - [(n-1)e],$$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$, and they can be generated quickly from the continued fraction expansion. If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in this article, which was published in Integers. This gives you a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac 1n \sum_{i=1}^n b_i \to e$. I don't have an answer for that. Yet.

As Gerry points out, the sequence $$a_n = [n e] - [(n-1)e],$$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$), and they can be generated quickly from the continued fraction expansion (of $e$, in this case). If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in this article, which was published in Integers. This gives you quickly a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac 1n \sum_{i=1}^n b_i \to e$. I don't have an answer for that. Yet.

As Gerry points out, the sequence $$a_n = [n e] - [(n-1)e],$$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$, and they can be generated quickly from the continued fraction expansion. If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in this article, which was published in Integers. This gives you a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac an \sum b_i \to e$. I don't have an answer for that. Yet.

As Gerry points out, the sequence $$a_n = [n e] - [(n-1)e],$$ where $[x]$ is the integer closest to $x$, has the desired extremal property. Unfortunately, one needs to know the value of $e$ to calculate the sequence in this way.

Fortunately, this is a typical example of a Sturmian sequence (on the alphabet $\{2,3\}$, and they can be generated quickly from the continued fraction expansion. If one uses the floor function in place of rounding, this has already been worked out by Ken Stolarsky and Tom Brown, and you can find a simple proof in this article, which was published in Integers. This gives you a large initial segment of the sequence; you cannot jump directly to $a_{1000000}$.

I haven't seen any detailed exposition using the "round" function (or ceiling function), but presumably it follows from the same principles.

A putman-ish followup question is to find a combinatorial process that generates a sequence $b_n$ with $\frac 1n \sum_{i=1}^n b_i \to e$. I don't have an answer for that. Yet.