More infomation on this continued fraction

Paul H. Erlich said:

Well, the frequency ratios of the noble "additive" or "logarithmic"
generators are 2^phi or 2^noble, which don't share the properties of the
noble ratios themselves. While the noble ratios have continued fraction
representations that end in all 1's, 2^phi, for example, has C.F.:

1 + 1/(1 + 1/(1 + 1/(6 + 1/(1 + 1/(2 + 1/(4 + 1/(1 + 1/(52 + 1/(2 + 1/(5 +
1/(4 + 1/(1 + 1/(106...)))))))))))))

Sarn says:

I find this use of nested parenthesis very hard to get my head around, and
it makes my eyes water.

What I would appreciate, and you might want to do for me Paul, is to place
this as an extended fraction, and give me an in depth description of how you
came up with this.

The reason I am intrigued by extended fraction, is because I chanced upon
one that generated the golden proportion as its convergence, and in addition
to this, I found that it converged IDENTICALLY, and at the same rate as the
Fibonacci series!!!

I have seen other expressions that generate phi, one with nested square
roots of one added to the square root of one added to the square root of one.

Also:

5^0.5*0.5+0.5 gives phi, and also:

2*cos(pi/5) radians give phi.

I fiddeled with my own extended fractions, and found what I feel is one that
gives the Tribonacci ratio, but the extended fraction above is surely
something new.

More information?

--Sarn.