Fibonacci Number
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
先从这个心智模型开始
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
它如何工作，不堆砌不必要的术语
Naive recursion recomputes the same subproblems and grows exponentially.
An iterative solution keeps only the previous two numbers, giving O(n) time and O(1) extra space.
把算法读成一连串决策
State the mathematical identity/property.
Choose numeric representation and constraints.
Translate the property into an iterative/recursive computation and analyze precision.
Read a tiny input by hand and write down the state after every important operation. For Fibonacci Number, the goal is to make the invariant visible rather than to memorize code.
function fibonacci(n) {
let a = 0, b = 1;
for (let i = 0; i < n; i++) {
[a, b] = [b, a + b];
}
return a;
}不要死记语法。跟踪每一行中发生变化的状态，并把它与上面描述的不变量联系起来。
O(n) iterative / O(2ⁿ) naive recursion
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
适用场景
- Use the direct mathematical property when it removes unnecessary brute force.
通常哪里会出错
- Ignoring floating-point precision or integer range.
- Using a formula outside its valid assumptions.
专业回答结构
Use Fibonacci to explain overlapping subproblems, memoization and the difference between elegant-looking recursion and efficient recursion.
What problem does Fibonacci Number solve?
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
关于复杂度应该提到什么？
Its typical cost here is O(n) iterative / O(2ⁿ) naive recursion, but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
最常见的错误是什么？
Ignoring floating-point precision or integer range.