%I #3 Feb 11 2014 19:05:41

%S 1,0,2,-1,8,-32,269,-3476,73030,-2482965,136563164,-12154121452,

%T 1750193489321,-407795083011416,153738746295304442,

%U -93780635240135708633,92561486982013944422368,-147820694710276269242519112,381968675131353879722669389589

%N a(1) = 1; a(n) = Fibonacci(n) - Fibonacci(n-1)* a(n-1) if n > 1.

%C 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.

%t t = {1}; gt = 1; For[i = 2, i <= 20, i++, gt = Fibonacci[i] - Fibonacci[i - 1] gt; t = Append[t, gt]]; t

%K sign

%O 1,3

%A _Joseph L. Pe_, Feb 20 2003