User blog comment:QuasarBooster/Fibonacci/Lucas Sequence Extentions?/@comment-2033667-20150730151541/@comment-24923514-20150730173737

That's very interesting! I've been used to seeing ratios like $$\frac{F_{n+1}}{F_n}$$ dip above and below φ, getting closer as n increases, so this Binet formula is a surprising but understandable answer. I also started tinkering around with sequences where the next entry is the sum of the previous three rather than two, such as $$S_a(n+1)=S_a(n-2)+S_a(n-1)+S_a(n)\mid S_a(0)=a,S_a(1)=1,S_a(2)=1$$. Would the Binet mathematics still be able to show φ as the limit for any of these possible seed combinations, as well as for any sequence in general where the next entry is the sum of an arbitrary number of previous entries?