The Thue-Morse Sequence (with visualizations)
In this video, we introduce the Prouhet-Thue-Morse sequence, which is a binary sequence. We discuss three methods to construct the sequence and then investigate some of the sequence's properties (including why it is the "fair sharing" sequence, the overlap-free property, its connection to a square-free sequence, how to use the sequence to build the Koch curve, it's connection to the Prouhet-Tarry-Escott problem, and an intriguing related infinite product representation of the square root of 2).
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The content for this video was taken from the amazing survey paper "The ubiquitous Prouhet-Thue-Morse sequence" by Jean-Paul Allouche and Jeffrey Shallit: https://cs.uwaterloo.ca/~shallit/Pape... . The paper includes many more facts of this truly remarkable sequence.
Here is a wonderful video from @standupmaths describing many of the same properties: • The Fairest Sharing Sequence Ever
#math #manim #sequence #integersequence #binary #thuemorse #thue #morse #fractal #kochcurve #roottwo #fairsharing
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