Given $f(0) = 1-c$, what is the non-recursive $f(n)$ that satify the following equation
$f(n)-\frac{1}{2}f(n-1)f(n)+f(n-1) = 1$


for n = 1,2,3,...?



What is the nonrecursive formula for the following implicit function?
$$ f(n) = 2{\frac {1-c+ \left( -1 \right) ^{n}-c \left( -1 \right) ^{n}-\sqrt 
{2}c+ \left( -1 \right) ^{n}\sqrt {2}c}{2-\sqrt {2}-\sqrt {2}c+2
 \left( -1 \right) ^{n}+ \left( -1 \right) ^{n}\sqrt {2}+ \left( -1
 \right) ^{n}\sqrt {2}c}} $$


which is a non-trivial pattern to spot! [I used a CAS] The thing to notice is that one can transform this first-order recurrence to the constant coefficient 2nd order linear recurrence
$$a(n+2)-2a(n) = 0, a(0) = 1, a(1) = -1-c$$
and then transform back. This can be done as the equation is of Riccati type.


I guess I 'cheated' in that I traced through the CAS's process of solving to extract the above information from it. Though if you don't know what you're looking for, it can be rather difficult information to extract...


