The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4.
If s(n) is rewritten in the form
s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms s(nk) and nk=n−k, what would that linear combination be and what would the minimum value of n have to be?
I do not know where to begin such a problem. I first started with reducing the given function, however that led to nowhere.
This problem was presented in a competition. What is the result, or what are some steps that could lead to a solution?
If $ s(n)=s(n−2)+s(n−3) $, then $ s(n-1)=s(n−3)+s(n−4) $ so $ s(n-3)=s(n−1)-s(n−4) $ so that $s(n) = s(n-2)+s(n-1)-s(n-4) =s(n-1)+s(n-2)-s(n-4) $.