Paired Fibonacci recurrence ($(F_{n+1},F_{n+2})=(F_{n+1},F_n+F_{n+1})$)

Compute adjacent Fibonacci numbers together, starting at $(0,1)$ and applying $(a,b)\mapsto (b,a+b)$. The shared pair avoids repeating the same recursive subproblem and gives $n$ additions for $F_n$. This counts arithmetic operations; exact integers grow with $n$.