math

TIL the sum of all fibs is the second-next fib minus one.

Kinda obvious once you think about it, but still... never seen this mentioned before.

I'd love to see this proof. Typically in math physics a lot of these proofs end up being totally wrong because they're assuming you can do things like distribute across an infinite series.

I would be shocked if this was actually analytically correct.

That makes sense. I haven't studied this in a long time but the statement on intuition gives me analytic combinatorics vibes. The statement on the relationships makes sense since structurally you kind of have to define the nth term with some n-k term.

What throws me off is the relationships with the "sum" of the sequence. It definitely feels weird when the sum is related to just an element of the series.

Proof by induction:

Base case: 1 = 2 - 1
Inductive case: if sum(fib[1...n]) = fib[n+2] - 1, then sum(fib[1...n+1]) = fib[n+1] + fib[n + 2] - 1 = fib[n + 3] - 1

On a more intuitive level, all exponentially growing sequences have a relationship between all terms up to a certain point and a single term beyond that point