A correct solution came from Will Cragoe and Bradley University alumnus Nathan Pauli. Other correct solutions were submitted by Nancy Scharzkopf, Lou Cairoli, A. Teitelman.
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If the sequence is allowed to have negative terms, there is no longest sequence, so let's consider Fibonacci-like sequences having only positive terms. The longest Fibonacci-like sequence then has ten positive terms. If the n'th term of the sequence, fn , is 2005 and fn-1 is x, then the values of the previous terms of the sequence are given in the the table on the left, together with the bound on x needed for fj to be a positive integer. The bound on fn-11 is inconsistent with the others, so the longest sequence has ten terms and x = 1239, namely, 25, 21, 46, 67, 113, 180, 293, 473, 766, 1239, 2005.
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