How to calculate how many steps are needed to solve the Nine Links

According to the gameplay of Nine Links, we can know that facing a state where all the rings are in the "down ring", if we want to go forward $n$ ring, then we must first go up first$n-1$ ring, then down before$n-2nd$ ring, then go up to$n$ ring, then go forward$n-2$ rings. Therefore, if we put the$The$ number of steps required for the$n$ loop is recorded as$a_n$, Then the sequence $a_n$Satisfy:

$$a_n=a_{n-1}+2a_{n-2}+1\text{\hspace{1em}(1)}$$

Here we think we can only operate one ring at a time, that is, we cannot move up and down the first two rings at the same time

Using this recurrence formula to write a python program, you can easily calculate $a_n=341$

def func(n):
    if n == 1:
        return 1
    elif n == 2:
        return 2
    else:
        return func(n-1) + 2*func(n-2)+1
res = func(9)
print(res)

So how to get $a_n$The general term formula of, after consulting the data extensively, we learned that $a_n$It belongs to the second-order linear constant coefficient non-homogeneous recurrence relationship. It can be calculated by formula.

The specific calculation method can be jumped to the reference materials at the end of the article

The final calculation result is
$$a_n=-\frac{1}{6}(-1{)}^n+\frac{2}{3}{2}^n-\frac{1}{2}$$

Also get $a_9=341$ Consistent with python calculation results.

---

Reference materials:

Baidu Library-2.4 Linear Constant Coefficient Inhomogeneous Recurrence Relation

CSDN-python implementation of Fibonacci sequence