[seqfan] Re: Decimal expansions related to Metallic Means, and Pascal Triangle

[William Keith]

Daniel Forgues' comments on this sequence seem most likely to be productive
for your investigation.

If we index the Fibonacci numbers like Mathematica, so that 1,1,2,... are
Fibonacci[1], Fibonacci[2], etc., then the generating function for the
Fibonacci numbers is

\sum_{i=1}^\infty Fibonacci[i] x^i = x/(1-x-x^2).

If x = 1/10^k, then this becomes

\sum_{i=1}^\infty Fibonacci[i] (1/10^k)^i = 10^k/(1- 1/10^k - 1/10^(2k)).

Letting k=1 and multiplying through by an additional 1/10 we get

\sum_{i=1}^\infty Fibonacci[i] / 10^(i+1) = (1/100)/(1- 1/10 - 1/100) =
1/89.

The generating functions for other Fibonacci-like sequences, such as the
Lucas numbers, have similar forms, although I don't recall them off the top
of my head.  What you are doing is evaluating these generating functions at
various x.

Certainly I think notes to this effect would make useful comments on any
related series.  Your question about "getting in to negatives" is probably
going to be clarified by assessing which generating functions and
evaluations your first steps are arising from.

Cordially,
William J. Keith