Yeah, it can definitely be optimised. I wanted to explain the thought process and focus less on how to make it faster, I think that's better for DP beginners.

according to you if k=5,dp[2]=15 but answer is 20 and dp[3]=45, but according to you dp[3]=30; please clear what you are storing in dp

I Tried Your Implementation , But it is giving wrong ans , try it for n=3, k=5 After 3rd year there should be 145 rats , i guess it is not considering the rats died after 3rd year , i have implemented the same , but dont know if it is correct or not , but it is passing all Samples TC

    int n,k;
    cin>>n>>k;

    vvi dp(n + 1, vi(3, 0)); // 2 means 2 days old , 1 means 1 day old

    dp[0][1] = k;

    rep(i,1,n+1)
    {

        dp[i][1] = dp[i - 1][1] * 2 + dp[i - 1][2] * 3;
        dp[i][2] = dp[i-1][1];
    }
    rep(i,0,n+1)
    {
        cout<<dp[i][1]<<" "<<dp[i][2] bn
    }

    vi rats(n+1,0);
    rats[0] = k;
    rep(i,1,n+1)
    {
        rats[i] = rats[i - 1] + dp[i - 1][1] * 2 + dp[i - 1][2] * 3;
        if(i>1)
            rats[i] -= dp[i - 2][2];
    }
    cout<<rats[n] bn

Hope it helps someone