The Sequence $$ a(1)=1,a(2)=1,a(3)=1,a(n+1)=a(n-2) a(n-1)+a(n) a(n-1)+a(n-2) a(n) $$
I was trying to find a formula for it.
After some failures, I searched it in OEIS and found A074047. It says, $a(n)$ tends towards $a(n-1)^\phi$ and $1.22376...^{\phi^n}$ where $\phi=(1+\sqrt{5})/2=1.6180339887...$
I am curious that how it could be derived, and how does the constant $1.22376...$ come ? . I am having no idea about it now.
Calculating the first few values of the sequence suggests that $\,a(n) \approx y\,x^{\phi^n}\,$ where $\,x>1.\,$ The appearance of $\,\phi\,$ comes the recursion equation $$ a(n+1) = a(n-2) a(n-1)+a(n) a(n-1)+a(n-2) a(n) $$ where the dominant terms are $\,a(n+1) = a(n)a(n-1)\,$ which is a multiplicative form of the Fibonacci recursion equation. Substituting the Ansatz into the recursion equation simplifying and removing common factors gives the equation $$ 1/y = x^{-\phi^{n+1}} + x^0 + x^{\phi^n-\phi^{n+1}}. $$ Letting $\,n\to\infty\,$ we find that $\,y=1 \,$ and $\, a(n)^{1/\phi^n} \to x \approx 1.22376155752637.\,$