FIBONACCI DECIMAL NUMBER PATTERNS VIA THE GENERATING FUNCTION

Russell Euler Northwest Missouri State University

The Fibonacci sequence is defined recursively by F_{1} = F_{2} = 1 and F_{n+2} = F_{n+1} + F_{n} for n ≥ 1. The first few terms of this sequence are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, · · ·. To six decimal places, the decimal expansion of 1/89 is .011235. Similarly,

\begin{matrix}\frac{1}{9899} & ≃ .0001010203050813213455 , and \\ \frac{1}{998999} ≃ & .000001001002003005008013021034055. \end{matrix}

Ignoring zeros, the occurrence of Fibonacci numbers in the above decimal expansions is apparent. If the decimal expansion of 1/89 is carried out to one more digit, the digit in the seventh decimal place is 9, not 8. The purpose of this paper is to explain these phenomena.

In [1, p. 3], the generating function for the Fibonacci sequence {F_{n}} was given to be

(1)  ∑_{n=1}^{∞} F_{n}x^{n} = \frac{x}{1 − x − x^{2}} .

The function defined by f(x) = \frac{x}{1−x−x^{2}} has singularities at (−1 ± \sqrt{5})/2. In magnitude, (−1 + \sqrt{5})/2 is the smallest of these two singularities. As a result, the interval of convergence of the series in (1) is |x| < (\sqrt{5} − 1)/2. Replacing x with 1/x in (1) leads to

∑_{n=1}^{∞} \frac{F_{n}}{x^{n}} = \frac{x}{x^{2} − x − 1} ,

and the region of convergence will be |x| > (1 + \sqrt{5})/2. So, in the region of convergence,

∑_{n=1}^{∞} \frac{F_{n}}{x^{n+1}} = \frac{1}{x^{2} − x − 1}.

Since x = 10^{k} is within the region of convergence for k ≥ 1,

\frac{1}{x^{2} − x − 1}|_{10^{k}} = ∑_{n=1}^{∞} \frac{F_{n}}{10^{k(n+1)}}

(2)  = \frac{1}{10^{2k}} + \frac{1}{10^{3k}} + \frac{2}{10^{4k}} + \frac{3}{10^{5k}} + \frac{5}{10^{6k}} + · · · .

In particular, if k = 1, then

\frac{1}{89} = \frac{1}{10^{2}} + \frac{1}{10^{3}} + \frac{2}{10^{4}} + \frac{3}{10^{5}} + \frac{5}{10^{6}} + ⋯
= .01 + .001 + .0002 + .00003 + .000005 + .0000008
+ .00000013 + ⋯
≃ .01123593.

Equation (2) explains why the Fibonacci numbers occur in the decimal expansions mentioned earlier — including the spacing of the digits with zeros. Furthermore, it is clear why the digit in the seventh decimal place of 1/89 is 9 instead of 8 — in the addition of the decimals in the terms of the series there is a ‘carry over’ of a 1 from the next term, 13, of the Fibonacci sequence.

\underline{Reference}

1. M. K. Azarian, “The Generating Function for the Fibonacci Sequence,” Missouri Journal of Mathematical Sciences, 2 (1990) 3–4.