[math-fun] Fibonacci, cosh, sinh
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.)
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.)
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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.)
Slightly better still: n 2 (- %i) sinh(n log(%i phi)) fib(n) = ----------------------------- sqrt(5) or even: n (- %i) sinh(n log(%i phi)) fib(n) = --------------------------- cosh(log(phi)) [with help from Rabinowitz] At 03:12 PM 2/29/2020, Henry Baker wrote:
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.)
Finally, 1-n %i sinh(n log(%i phi)) fib(n) = ------------------------- sinh(log(%i phi)) At 10:53 PM 2/29/2020, Henry Baker wrote:
Slightly better still:
n 2 (- %i) sinh(n log(%i phi)) fib(n) = ----------------------------- sqrt(5)
or even: n (- %i) sinh(n log(%i phi)) fib(n) = --------------------------- cosh(log(phi))
[with help from Rabinowitz]
At 03:12 PM 2/29/2020, Henry Baker wrote:
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.)
participants (2)
-
Henry Baker -
Tom Karzes