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3 Suppose 0 < (3 < 1/2 is irrational and let 7 be the unsigned distance from 1/(3 to the nearest integer. Let j = \l/(3\. Then (1) The first j entries of sn((3) are the word C consisting of n — 1 O's followed by a single 1; (2) If 0 < {1/(3} < 1/2 then the rest of sn{(3) after the initial word C consists of strings of the word A = 0 J _ 1 1 broken by single instances of the word B = CP1, and the rth such A or B is A if sr("f) = 0, B if s r('j) = 1/ omd (3) If 1/2 < {1//3} < 1 then A = (PI and B = 0 J ' _1 1 but the rest is the same.

It need not be the case that 0[a] = Z[a], that is, there may be integers of 0[a] not of the form ^Zo _ 1 akakAn ideal in 0[a] is an additive subgroup I of 0[a] that is closed under multiplication by O[o\. That is, if /? e I and 7 € 0[a\ then @j £ I. The set 0[a] is itself an ideal. It is the largest ideal of 0[a], while {0} is the smallest. Every ideal of 0[a] is finitely generated. i,... ,j3m) so that every element of I has the form J2T "fkPk for some integers 7fc of 0[a\. Every ideal also has an integral basis.

While d' < d, (and while Length[C]< b if a = a/b is rational), execute the following loop: Let $ = a1 - [a' + 1/2J. Let j = [1//3J • Update A and B to Aj~1B and AjB respectively if {1//3} < 1/2, or to AjB and A^lB if {1/(3} > 1/2. Update C to CA^^B. Update a' to 1//3. Increment d'. Output C,a',A, and B. 1 There is a recreational mathematics side to this. Larry Tessler has given a fetching metaphor: "A tank rolls forever along a perfectly straight infinite highway whose center line is studded with reflectors spaced exactly 1 meter apart.