The 0110 sequence consists of an item followed by its opposite.

Take

0as the initial item, and1as its opposite.

So the sequence starts out## 0 1

To continue the sequence, take the whole sequence so far and use

itas the "item". The "opposite' of an item means switching a0to a1, and a1to a0. So the opposite of01is10.The continued sequence is:

0110The opposite of

0110is1001, so the sequence continues as:## 0110 1001

Taking this sequence as an item, the sequence continues as:

## 0110 1001 1001 0110

Taking this sequence as an item, the sequence continues as:## 0110100110010110 1001011001101001

Taking this sequence as an item, the sequence continues as:## 0110100110010110 1001011001101001 1001011001101001 0110100110010110

and so on...

Use your left index finger for

0and your right index finger for1.

Or, use an index finger and a middle finger on the same hand.

Or sit at a computer and type two keys in this pattern.

**Having done this for a long time**, I can tap out the first 256 digits
consistently.

**There are never** more than two **1**s or two **0**s in a row.

**Does this sequence ever repeat?** That is, is the number represented by
a decimal point followed by the digits in this sequence a **rational number**?

This sequence is known as the Thue-Morse Sequence, and not only is it irrational, it has been proven to be transcendental.

What fascinates me about this sequence is that it contradicts my simplistic notion that irrational numbers don't have a pattern. This sequence has a very simple pattern -- it is hard to imagine a simpler pattern than an item followed by its opposite -- and yet it is irrational. The pattern is powerful because it is a meta-level pattern.

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