Cluster algebras
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What do the following facts have in common?
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 * The recurrence defined by
 x(n+1)x(n-1) = x(n) + 1
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   is periodic with period 5 for arbitrary starting values of x(0) and x(1).
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 * The number of tilings by dominos of an n'th order Aztec Diamond
   (the region obtained from four staircase shapes of height n by
   gluing them together along the straight edges) is 2^(n(n+1)/2).
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 * Ptolemy's theorem for quadrilaterals inscribed in a circle: if A,
   B, C, D are the lengths of the edges of a cyclic quadrilateral, and
   E and F are the lengths of the diagonals, then EF = AC + BD.
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The answer is that they are all instances of cluster algebras defined
by Fomin and Zelevinsky, a developing story in combinatorics with
a striking variety of connections.