David Brooks sent me a note recently with the following interesting tidbits on the sum of the digits of some Fibonacci numbers. Thought it was fun, so here they are. Thanks David:

btw: here is OEIS sequence

Find the next one, and send it here...

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I ran across this interesting information while researching information on Leon Bankoff.

The sum of the digits of the 5

^{th}Fibonacci number is 5. (Leon Bankoff) (Ok, the F(5) is five, so that one is a gimme'
The sum of the digits of the 10

^{th}Fibonacci number (55) is 10. (Leon Bankoff)
The sum of the digits of the 31

^{st}Fibonacci number is 31 (`1346269)`. (Leon Bankoff)
The sum of the digits of the 35

^{th}Fibonacci number is 35. (Leon Bankoff)
The sum of the digits of the 62

^{nd}Fibonacci number is 62. (Leon Bankoff)
The sum of the digits of the 72

^{nd}Fibonacci number is 72. (Leon Bankoff)
The sum of the digits of the 175

^{th}Fibonacci number is 175. (OEIS)
The sum of the digits of the 180

^{th}Fibonacci number is 180. (OEIS)
The sum of the digits of the 216

^{th}Fibonacci number is 216. (OEIS)
The sum of the digits of the 251

^{st}Fibonacci number is 251. (OEIS)
The sum of the digits of the 252

^{nd}Fibonacci number is 252. (OEIS)
The sum of the digits of the 360

^{th}Fibonacci number is 360. (OEIS)
The sum of the digits of the 494

^{th}Fibonacci number is 494. (OEIS)
The sum of the digits of the 504

^{th}Fibonacci number is 504. (OEIS)
The sum of the digits of the 540

^{th}Fibonacci number is 540. (OEIS)
The sum of the digits of the 946

^{th}Fibonacci number is 946. (OEIS)
The sum of the digits of the 1188

^{th}Fibonacci number is 1188. (OEIS)
The sum of the digits of the 2222

^{nd}Fibonacci number is 2222. (OEIS)
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More mature students should read Joshua Zucker's comment below to begin to understand a little more about the frequency with which these occur.

Younger students might think about the digital root of the Fibonacci numbers since that only requires the sum of two simple one digit numbers.

They might want to explore questions like:

How frequently is the digital root of F(n) the same as the digital root of n (obviously in all the case above, but are there others?)

There be a point where the sequence of digital roots repeats; Why?

How would that information help you find large numbers for which the digital root of F(n) is the same as the digital root of n?

Can you see how this would provide a shortcut in the search for numbers in like the sequence David Brooks has written about above?

What is the longest possible string length of one digit numbers before a repeating sequence must emerge? (Perhaps you might attempt that question with only two digits, zero and one, or three digits and then expand the question).

There are some other interesting things to explore, for instance the sum of the digits of F(2314) is 2134.

More mature students should read Joshua Zucker's comment below to begin to understand a little more about the frequency with which these occur.

Younger students might think about the digital root of the Fibonacci numbers since that only requires the sum of two simple one digit numbers.

They might want to explore questions like:

How frequently is the digital root of F(n) the same as the digital root of n (obviously in all the case above, but are there others?)

There be a point where the sequence of digital roots repeats; Why?

How would that information help you find large numbers for which the digital root of F(n) is the same as the digital root of n?

Can you see how this would provide a shortcut in the search for numbers in like the sequence David Brooks has written about above?

What is the longest possible string length of one digit numbers before a repeating sequence must emerge? (Perhaps you might attempt that question with only two digits, zero and one, or three digits and then expand the question).

There are some other interesting things to explore, for instance the sum of the digits of F(2314) is 2134.

## 1 comment:

This is interesting! I never knew before that log(phi) was so close to 0.2. That explains why there are so many of these things: with the nth Fibonacci number being phi^n and thus having a log of 0.2n, it has about 0.2n digits, which average 5 or so, so the digit sum is about n.

I suppose the actual average digit is more like 4.5, (10% too small) so the digit sum is a little too small, which in the long run probably kills it (I mean, wouldn't you be surprised if there were a trillion-digit Fibonacci number whose digits were on average as big as 5 instead of being 4.5?).

Oh, but wait, log(phi) is about 4% bigger than 0.2, so this is even closer than I thought; we only need an average digit of about 4.8 instead of 4.5.

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