General Fibonacci summation sequence failure

[Don86326]

Hello!  1st post... really nervous, because I am "et up" with academic elitism and don't want hero's worshipped remotely bursting my adoration with harsh retort to my naivete.  Consider I'm six years old. (Actually 67, retired from software development, but work with me.)

I found an anomaly in general summation, Fibonacci style -> F(n) = F(n-1) + F(n- 2)

Algebra and computer science was fun, and the fun went to amazement when two seed numbers for general Fibonacci summation produced adjacent sequence numbers with a divisor that is not (it is not) an approximation of the golden ratio.  The quotient of adjacent sequence numbers is -1/Phi = -0.61803... [Big, bold, oversized exclamation point goes here.  A red one.]

Moreover, the sequence does not exponentially increase, but rather oscillates toward zero.

Did I stumble into something everyone knows about in algebra?

Try this summation beginning with two seeds of... (Phi = 5^0.5 * 0.5 + 0.5 = golden ratio, computer code style)

Seed #1 = -Phi^(N)  = (A negative number)

Seed #2 - +Phi^(N-1)   = (A positive number)

I assume the seed numbers create a Fibonacci identity that propagates the effect.  Perhaps every addition flips the identity in sign, but as another Fibonacci degree preserving the identity.

An identity conveyor.

Notice that on any machine this is done (spreadsheet or supercomputer) that as the sign-oscillation of the resulting sequence approaches zero, one of the values will reach the machine-zero (limit of floating point digits) before the other value, and this begins a new normal exponentially increasing summation sequence of 0 and some small number.  This will appear on the sequence graph as a impulse with two peaks.  The impulse is actually a positive and negative pair of asymptotes to infinity, but a graph will often map +/- infinity as a double blip.

[swansont]

!

Moderator Note

Moved, because this doesn’t seem to be a puzzle, as such.