| Fibonacci Statue in Pisa, *plus.maths.org |
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181... Ok, You get the idea...
I just came across a couple of odd references to things that produce Fibonacci like sequences, at least for a short while...
It started when I posted a note on the 89th day of the year (You do the math) that 89 is the fifth Fibonacci prime and the reciprocal of 89 starts out 0.011235... (generating the first five Fibonacci numbers) (which I found at the Prime Curios page. (Ok footnote here, I didn't realize until just recently that if the nth Fibonacci number is prime, then n is also prime, with one exception. but not the converse)
Don S McDonald (@McDONewt) then sent me another... 69 Choose 5 = 1 1 2 3 8 5 13 which is so nearly perfect...
Now I'm thinking there must be more of these... so fire away... Send them in and I'll make a collection we can all share..
In the comments, Joshua Zucker pointed out that the decimal expansion of 1/89 goes well beyond 0,1,1,2,3,5.. The next digit is nine, instead of 8, because it includes the tens digit fromthe 13 that would follow, and the following digit is the sum of the 3 from 13, and the 2 from 21, and you could continue this way indexing the next Fibonacci term one to get more. When Joshua said all the digits, he didn't mean ought to thousands of digits, since the period of the fraction is only 44 digits.
2 comments:
1/89 generates ALL the fibonacci numbers, it's just that carrying messes them up.
Try 1/9899 or 1/998999 or ... you get the idea.
Really what we're doing here is expanding 1/(x^2 - x - 1), which generates the Fibonacci sequence just fine, but when we work in base 10 instead of base x, we run into trouble with the carrying. So, do it in base 100, or base 1000, or ...
I want Joshua Zucker's brain when he is through with it.
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