This will be a kind of(?) short article; just wanted to have some discussion over this finding/curiosity.
I know how familiar the people of this site are with the concept of recursion, and I'm sure a fair amount of you have heard about the primitive-recursive Fibonacci series, in which the first two numbers are given as 0 and 1, or 1 and 1 (as they yield the same result). Further numbers in the sequence are, simply enough, the summation of the two previous entries (1+1=2, 1+2=3, 2+3=5, etc). I'd imagine the sequence's growth rate is absolutely puny, so I'll be quick.

An interesting number emerges as the limit of ratios between consecutive numbers from sufficiently high places in the sequence. The value is approximately 1.618, and is known as phi, φ (cough cough my favorite number cough). The Lucas Sequence begins instead with 2 and 1, and it too approaches φ for large numbers in its series. Interestingly, respective powers of φ and numbers in this Lucas series closely resemble one another.

The Googological stuff

Although this extention doesn't produce very large numbers, it could possibly be considered relevant in that its growth rate can be analyzed (hopefully that makes it enough). I'll cut to the chase: I thought to generalize the two aforementioned sequences to generate a new diversity of numbers, as well as to see if anything comes out after pumping it with Googological "madness" (ongoing). \[S_a(n+1)=S_a(n-1)+S_a(n)\mid S_a(0)=a,S_a(1)=1\] An example series is {3,1,4,5,9,14...}, where a=3. The Fibonacci and Lucas sequences can be expressed as such: \[S_0(n)\Rightarrow S_1(n)=F_n,S_2(n)=L_n\] I wonder myself about the growth rates of these sequences... Anyway, one will find the emergence of φ in any of these series by dividing higher numbers within them. That is, of course, unless one considers the series \(S_\omega(n)\), or rather \(S_n(n)\).

I can't say too much about this series (0,1,3,5,11,20,38,69...), considering how much I'm not a PhD mathematician, but as it is a diagonalization, I speculate that it has a different growth rate altogether than any \(S_a(n)\).

Also, what I find most intriguing; when one evaluates the ratios of consecutive numbers farther and farther through this series, the pervasive 1.618... is not what seems to come out! I have limited computational power, but the ratio I found so far approached 1.65... I'm very curious about this potentially new limit, so I encourage the investigation and discussion by anyone also interested in what I've spoken about.

I really hope this was relevant enough to the wiki XD

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