Patterns Starting With 4
Patterns that can be factored: \[[4^*04^*_{10}]_n=4\cdot{10\cdot10^{2n}-9\cdot10^{n}-1\over9}=4\cdot\left(10^{n}-1\right)\left(10\cdot10^{n}+1\right)/9=4\cdot\left(9\cdot[1^*_{10}]_n\right)\left([10^*1_{10}]_n\right)/9\] \[[4^*84^*_{10}]_n=4\cdot{10\cdot10^{2n}+9\cdot10^{n}-1\over9}=4\cdot\left(10^{n}+1\right)\left(10\cdot10^{n}-1\right)/9=4\cdot\left([10^*1_{10}]_{n-1}\right)\left(9\cdot[1^*_{10}]_{n+1}\right)/9\] \[[4^*14^*_{10}]_n={40\cdot10^{2n}-27\cdot10^{n}-4\over9}=\left(5\cdot10^{n}-4\right)\left(8\cdot10^{n}+1\right)/9=\left([9^*2_{10}]_n/2\right)\left(9\cdot[8^*9_{10}]_{n-1}\right)/9\] \[[4^*74^*_{10}]_n={40\cdot10^{2n}+27\cdot10^{n}-4\over9}=\left(5\cdot10^{n}+4\right)\left(8\cdot10^{n}-1\right)/9=\left(9\cdot[1^*2_{10}]_n/2\right)\left([79^*_{10}]_n\right)/9\]