[c++] simple graph + memoization  based solution with comments
\nclass Solution {\npublic:\n    \n    vector<vector<long long int>>dp;\n    vector<string>v;\n    #define mod 1000000007\n    \n    //generating all proper str
```\nclass Solution {\npublic:\n    \n    vector<vector<long long int>>dp;\n    vector<string>v;\n    #define mod 1000000007\n    \n    //generating all proper strings (no adj of same color of size m )\n    \n    void recurs(int m,char prev,string s)\n    {\n          if(m==0)\n            {\n            v.push_back(s);\n            return ;\n            }\n        \n        string c="RGB";\n        \n        for(auto x:c)\n        {\n            if(x!=prev)\n                recurs(m-1,x,s+x);\n        }\n    }\n    \n    \n    long long  int recurs(vector<int>graph[],int i,int n)\n    {\n        if(n==1)\n             return 1;\n        \n        if(dp[i][n]!=-1)\n             return dp[i][n];\n        \n        long long int a=0;\n        \n        for(auto x:graph[i])\n            a+=recurs(graph,x,n-1)%mod;\n        \n        return dp[i][n]=a%mod;\n    }\n    \n    int colorTheGrid(int m, int n) {\n        \n        //generating valid strings\n        recurs(m,\'@\',"");\n        \n        \n        // for(auto x:v)\n        //     cout<<x<<" ";\n        \n        vector<int>graph[v.size()];\n        \n        //graph construction\n        \n        for(int i=0;i<v.size();i++)\n        {\n            for(int j=0;j<v.size();j++)\n            {\n                bool flag=1;\n                for(int k=0;k<m;k++)\n                {\n                    if(v[i][k]==v[j][k])\n                         {\n                        flag=0;\n                        break;\n                         }\n                }\n                if(flag&&i!=j)\n                {\n                    graph[i].push_back(j);\n                }\n            }\n        }\n        \n        \n        //finding all valid paths from all possible startings\n        \n        long long int ans=0;\n        dp.resize(v.size(),vector<long long int>(n+1,-1));\n        \n        for(int i=0;i<v.size();i++)\n            ans+=recurs(graph,i,n)%mod;\n\n        return ans%mod;\n        \n    }\n};\n```
['Dynamic Programming', 'Graph', 'C']