ASKSAGE: Sage Q&A Forum - Latest question feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 20 Nov 2019 10:33:55 -0600How to find a short form of recursive defined sequences?https://ask.sagemath.org/question/48820/how-to-find-a-short-form-of-recursive-defined-sequences/Hi, I'm new to sagemath.
Is there any way to so calculate/solve/find a short version of a recursive defined sequence?
E.g. I have a sequence like: (Fibonacci)
def f(n):
if n == 0:
return 0
if n == 1:
return 1
if n == 2:
return 1
else:
return f(n-1)+f(n-2)
How can I compute a short form of $f_n$?
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In this example case $f_n$ would be:
$f_n=\frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^n - \frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$
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Edit:
Thanks to Emmanuel I found how to solve those equations in pdf:
from sympy import Function,rsolve
from sympy.abc import n
u = Function('u')
f = u(n-1)+u(n-2)-u(n)
rsolve(f, u(n), {u(0):0,u(1):1})
-sqrt(5)*(1/2 - sqrt(5)/2)**n/5 + sqrt(5)*(1/2 + sqrt(5)/2)**n/5maanWed, 20 Nov 2019 10:33:55 -0600https://ask.sagemath.org/question/48820/