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then the dynamics of any of , , and can be expressed as the dynamics of in

:

which is patterned on the characteristic equation of .

Moreover, an infinitude of other third-order univariate difference equations can be found by multiplying any of the three matrices together an arbitrary number of times in an arbitrary sequence. For instance, the matrix moves one out the tree by two nodes (across, then down) in a single step; the characteristic equation of provides the pattern

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