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Problem of the Week 873

A Continued Monster

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Assume that the monster fraction shown below is a well-defined real number. Then it is a very familiar number. What is it?

                1 + ...
            1 + -------- 
                3 + ...
        1 + ------------
                5 + ...
            3 + --------
                1 + ...
    1 + ----------------
                1 + ...
            5 + --------
                3 + ...
        3 + ------------
                5 + ...
            1 + --------
                1 + ...
1 + --------------------
                1 + ...
            1 + --------
                3 + ...
        5 + ------------
                5 + ...
            3 + --------
                1 + ...
    3 + ----------------
                1 + ...
            5 + --------
                3 + ...
        1 + ------------
                5 + ...
            1 + --------
                1 + ...

The pattern is that the columns, from the top, come from:

Source: Recent Crux Math., which refers to Mathematics and Informatics Quarterly 1/97 vol. 7, which refers to an original contribution by Edwin O. Buchman (Fullerton, CA) Jan/Feb 1996 Quantum (which has 3 such weird continued things). © Copyright 1998 Stan Wagon. Reproduced with permission.

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3 November 1998