Looking at a slight variation of the fibonacci sequence
f(x) = f(x-1) + f(x-2) + 1
where f(1) = 1, f(2) = 1

I'm trying to find the ratio of this sequence but can't figure out how. To get an approximation I just tried looking at some random examples, (i.e. f(10)/f(9), f(20)/f(19))
And it seems that this is the golden ratio, but I can't understand how that is correct as this sequence seems to grow much faster than the fibonacci sequence

  • 1
    $\begingroup$ We can tell that if this modification of the Fibonacci sequence tends to a geometric sequence (as the Fibonacci sequence itself does), then the common ratio must also be $\varphi$, since if we divide by $f(x-2)$ and take the limit, we get $r^2 = r + 1$ (since $1/f(x-2) \to 0$), which is the same characteristic equation as the Fibonacci series has. $\endgroup$
    – Brian Tung
    Apr 23 '20 at 20:15

If you write $g(x)= f(x)+1$ you got $$g(x+1)=g(x)+g(x-1)$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.