Fibonacciness
David W. Wilson
wilson at cabletron.com
Mon Apr 20 18:11:39 CEST 1998
For a, b > 0, define the "reverse Fibonacci sequence" R(a, b) as
follows:
R(0) = a; R(1) = b; R(n+2) = R(n) - R(n-1).
For instance, R(20, 11) is
20, 11, 9, 2, 7, -5, 12, -17, ...
Now, define f(a, b) to be the element of R(a, b) immediately that
preceeds the first nonpositive element. Thus f(20, 11) = 7.
Finally, define f(a), the Fibonacciness of a, to be the minimum of
f(a, b) over all b >= 1. The smaller that f(a) is, the more
Fibonaccilike a is.
Questions:
Let c = (sqrt(5)-1)/2 = .6180339...
Does f(a) = min(f(a, floor(ca)), f(a, ceil(ca)))?
Which numbers have Fibonacciness 1? 2? 3?
What is the smallest number of Fibonacciness n (the answer is
an interesting sequence)?
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