Problem  of the  Week

Find an ordered list of positive integers, a₁, a₂,..., with the fewest possible integers, satisfying all the following properties:

(a) Every positive integer is the sum of numbers from the list,
(b) no number on the list appears more than once in any one sum, and
(c) no two consecutive numbers on the list, that is, a_k, a_k+1, appear in any one sum.

(Note that an integer is considered to be the sum of one number on the list if it is actually on the list. Also, for the terminally picky, "fewest possible" refers to inclusion, not to cardinality.)

In symbols, for any positive integer n

where the sum runs over all elements of the list, b_i = 0 or 1, and b_i b_i+1 = 0.

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