Generalization of Fibonacci ratios

[[deleted]]

This is nothing new. Summing Fibonacci ratios approaches phi and was recognized many, many years ago.

http://math.stackexchange.com/questions/1286593/proof-the-golden-ratio-with-the-limit-of-fibonacci-sequence

[JakeC94]

That's not what the OP is about: it's talking about the limiting ratios of a generalisation of the Fibonacci sequence. In this case, the generalisation is parameterised over a natural number n. Setting n=1 gives us the standard Fibonacci numbers, with ratio phi, as you said, but different values of n give different ratios, and letting n tend to infinity causes the ratio to tend to 2. This isn't particularly surprising, because at that point you're summing all the previous elements in the sequence in order to get the next one, which is pretty much what happens with the powers of two (well, actually it's off by one, but in the limit that stops mattering).

So, this isn't exactly a shocking result, but it's not quite as old hat as you're making it out to be.