Generating Functions
Here are some neat generating functions for different notable sequences of numbers.
Episode 3: All Horses Are the Same Color -- Equine Monochromaticity [#MathChops]
Obviously, this theorem is false, but it is a good way to show off your math chops and confuse a friend who may be taking an introductory course in math reasoning. This ‘proof’ is purely for fun, but does point out an important part of inductive proofs, which is that the assumption for the ‘n’th case must imply our statement is true in the ‘n+1’th case for any arbitrary n. Take what you will from this proof, but it reminds me of a joke I heard once. A mathematician, physicist, and engineer are on a train in spain and see a white horse. The engineer remarks, “all horses are white!” to which the physicist and mathematician shake their heads. “No no no,” says the physicist, “what this means is that some horses in spain are white.” to which the mathematician shakes his head. The mathematician thinks for a little, and says “In passing we saw a white horse grazing in the plains of spain; therefore, there exists at least one horse in spain, of which at least one side is white.” and the thre...
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