and more fun
on generating interesting sequences with simple fractions. (via Science News)
an excerpt
Calculate 100/89. You get the decimal expansion 1.1235955056 . . .
Look closely, and you'll see that this fraction generates the first five Fibonacci numbers (1, 1, 2, 3, and 5) before blurring into other digits. Recall that, starting with 1 and 1, each successive Fibonacci number is the sum of the two previous Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on.
Calculate 10000/9899. This time, you get 1.0102030508132134559046368 . . .
This fraction generates the first 10 Fibonacci numbers (using two digits per number). Going further, the fraction 1000000/998999 generates the first 15 Fibonacci numbers (using three digits per number).
Note that, in successive fractions, two 0s are appended to the numerator and a 9 to the beginning and end of the denominator.
Will the next fraction, 100000000/99989999, generate the first 20 Fibonacci numbers? Does the pattern continue forever? The answer appears to be yes.