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What's the explanation of the Fibonacci sequence appearing in the result of 1/89, as demonstrated by http://www.goldennumber.net/Number89.htm and shown below? If you wish, also explain the relation to the number 109 too.

1 / 89 = 0 / (10 ^ 1 ) + 1 / (10 ^ 2 ) + 1 / (10 ^ 3 ) + 2 / (10 ^ 4 ) + 3 / (10 ^ 5 ) + 5 / (10 ^ 6 ) + 8 / (10 ^ 7 ) + 13 / (10 ^ 8 ) + ...  0.011235955... = 0.0 + 0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005 + 0.0000008 + 0.00000013 + ... 

(This question was inspired by What is special about the numbers 9801, 998001, 99980001 ..?.)

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    The keyword here is "generating function." I might add a more detailed response later.2012-01-26
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    @QiaochuYuan: Sorry, didn't see your comment. Please feel free to add your answer.2012-01-26
  • 1
    See also [this question](http://math.stackexchange.com/questions/31085/for-which-number-does-multiplying-it-by-99-add-a-1-to-each-end-of-its-decimal-re).2012-01-26

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