# [seqfan] Re: Any other thoughts, opinions on (in)finiteness of A020995?

Charles Greathouse charles.greathouse at case.edu
Mon Mar 19 18:34:20 CET 2012

For clarification: I'm quite sure it's finite, but my calculations
below seem to have some mistake which make the numbers too low.  When
adjusted the results should give some confidence as to whether all
terms are known, but in any case it should be finite since the value
should be 0.94...n + O(n^(0.5 + o(1))).

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Mar 19, 2012 at 1:25 PM, Alonso Del Arte
<alonso.delarte at gmail.com> wrote:
> Long ago, Robert G. Wilson v suggested that the sequence of n such that the
> sum of the base 10 digits of Fibonacci(n) is equal to n (A020995) might be
> infinite, though only twenty terms are known and the largest known term is
> quite small. In 2006, Stefan quoted an argument from Robert Dawson that not
> only is the sequence likely finite, we might already know all the terms.
>
> Most recently, Charles posted the following argument in a comment for the
> Sequence of the Day for April 20, saying that it needs to be checked:
>
> "The number of digits in the n-th Fibonacci number is
> $n\log_{10}\varphi+o(1)$, so the expected digit sum is about
> 0.94n.  Modeling the result as a normal distribution, the variance is about
> $82.5\log_{10}\varphi$ and so the heuristic probability that the
> digits of $F_n$ are large enough to be in the sequence is about
> $\operatorname{erfc}\left(n\frac{1-4.5\log_{10}\varphi}{\sqrt{165}}\right)/2.$
> This decays rapidly: by n = 10,000 it is below $10^{-935}.$"
>
> Have you any other thoughts, opinions, those of you who have pondered this
> or similar questions?
>
> Al
>
> --
> Alonso del Arte
> Author at SmashWords.com<https://www.smashwords.com/profile/view/AlonsoDelarte>
> Musician at ReverbNation.com <http://www.reverbnation.com/alonsodelarte>
>
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