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Fixed minor typos and notation
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Gordon Royle
Gordon Royle
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Here is a small visualisation of the values of this function.

I have taken $n = \{1,2,\ldots,512\}$$n \in \{1,2,\ldots,512\}$ and calculated the proportion of the numbers $0 \leqslant i < 2^n$$0 \leqslant x < 2^n$ that have a maximum 1-run of even length.

enter image description here

Obviously I can't compute directly the longest run for each of the integers smaller than $2^{512}$ but luckily the values can be computed symbolically.

The number of compositions (i.e., ordered partitions) of $n$ with largest part odd is equal to the number of integers in the range $0 \leqslant x < 2^{n-1}$ with largest 1-run of even length (via the "McMahon Graph" of a composition).

These values are listed in the OEIS at https://oeis.org/A103421 along with a generating function that can be computed symbolically by MathematicsMathematica for a reasonable range of values.

Here is a small visualisation of the values of this function.

I have taken $n = \{1,2,\ldots,512\}$ and calculated the proportion of the numbers $0 \leqslant i < 2^n$ that have a maximum 1-run of even length.

enter image description here

Obviously I can't compute directly the longest run for each of the integers smaller than $2^{512}$ but luckily the values can be computed symbolically.

The number of compositions (i.e., ordered partitions) of $n$ with largest part odd is equal to the number of integers in the range $0 \leqslant x < 2^{n-1}$ with largest 1-run of even length (via the "McMahon Graph" of a composition).

These values are listed in the OEIS at https://oeis.org/A103421 along with a generating function that can be computed symbolically by Mathematics for a reasonable range of values.

Here is a small visualisation of the values of this function.

I have taken $n \in \{1,2,\ldots,512\}$ and calculated the proportion of the numbers $0 \leqslant x < 2^n$ that have a maximum 1-run of even length.

enter image description here

Obviously I can't compute directly the longest run for each of the integers smaller than $2^{512}$ but luckily the values can be computed symbolically.

The number of compositions (i.e., ordered partitions) of $n$ with largest part odd is equal to the number of integers in the range $0 \leqslant x < 2^{n-1}$ with largest 1-run of even length (via the "McMahon Graph" of a composition).

These values are listed in the OEIS at https://oeis.org/A103421 along with a generating function that can be computed symbolically by Mathematica for a reasonable range of values.

Source Link
Gordon Royle
Gordon Royle
  • 12.7k
  • 1
  • 51
  • 73

Here is a small visualisation of the values of this function.

I have taken $n = \{1,2,\ldots,512\}$ and calculated the proportion of the numbers $0 \leqslant i < 2^n$ that have a maximum 1-run of even length.

enter image description here

Obviously I can't compute directly the longest run for each of the integers smaller than $2^{512}$ but luckily the values can be computed symbolically.

The number of compositions (i.e., ordered partitions) of $n$ with largest part odd is equal to the number of integers in the range $0 \leqslant x < 2^{n-1}$ with largest 1-run of even length (via the "McMahon Graph" of a composition).

These values are listed in the OEIS at https://oeis.org/A103421 along with a generating function that can be computed symbolically by Mathematics for a reasonable range of values.