Yes! Thank you! I also get: n n (2 log(phi) + %i %pi) 2 (- %i) sinh(-----------------------) 2 fib(n) = --------------------------------------- sqrt(5) where it's obvious that fib(0)=0. At 02:59 PM 2/29/2020, Tom Karzes wrote:
Let:
A = 1/sqrt(5) B = -1/sqrt(5) a = log(phi) b = -log(phi) + i*pi
Then:
fib(n) = A*exp(a*n) + B*exp(b*n)
This makes use of the fact that:
exp(i*pi*n) = (-1)^n
for integer n.
Tom
Henry Baker writes:
It is well-known that
fib(n) = cosh(n*alpha)/cosh(alpha) (n odd) fib(n) = sinh(n*alpha)/cosh(alpha) (n even)
alpha = asinh(1/2).
Also,
fib(n) = phi^n-(-phi)^(-n) ----------------- sqrt(5)
Is there any version of these equations
fib(n) = A*exp(a*n)+B*exp(b*n)+C*exp(c*n)+...
that *doesn't* require splitting up into n odd, n even?
(I'm willing to accept complex A,a,B,b,C,c, etc.)