OFFSET
1,3
COMMENTS
1. Let s(n) be a sequence such that lim s(n)/s(n+1) = K different from -1. The "oscillator sequence" (or simply "oscillator") of s(n) is the sequence s'(n) defined by the rules: s'(1) = 1; s'(n) = 1 - (s(n-1)/s(n)) s'(n-1). 2. It is an open problem whether the oscillator (prime)' converges to 1/2 or diverges. 3. s'(n) = 1 - (s(n-1)/s(n)) s'(n-1) = [s(n) - s(n-1) s'(n-1)]/s(n). The numerator is the expression s(n) - s(n-1) s'(n-1), which motivates the definition of the above sequence a(n). a(n) is called the "integral oscillator" of Fibonacci(n). In general the integral oscillator of s(n) can be defined similarly.
MATHEMATICA
t = {1}; gt = 1; For[i = 2, i <= 20, i++, gt = Fibonacci[i] - Fibonacci[i - 1] gt; t = Append[t, gt]]; t
CROSSREFS
KEYWORD
sign
AUTHOR
Joseph L. Pe, Feb 20 2003
STATUS
approved