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 ..?.)