Patterns Starting With 6
Patterns that can be factored: \[[6^*06^*_{10}]_n=2\cdot{10\cdot10^{2n}-9\cdot10^{n}-1\over3}=2\cdot\left(10^{n}-1\right)\left(10\cdot10^{n}+1\right)/3=2\cdot\left(9\cdot[1^*_{10}]_n\right)\left([10^*1_{10}]_n\right)/3\] \[[6^*46^*_{10}]_n=2\cdot{10\cdot10^{2n}-3\cdot10^{n}-1\over3}=2\cdot\left(2\cdot10^{n}-1\right)\left(5\cdot10^{n}+1\right)/3=2\cdot\left([19^*_{10}]_n\right)\left(3\cdot[16^*7_{10}]_{n-1}\right)/3\] \[[6^*86^*_{10}]_n=2\cdot{10\cdot10^{2n}+3\cdot10^{n}-1\over3}=2\cdot\left(2\cdot10^{n}+1\right)\left(5\cdot10^{n}-1\right)/3=2\cdot\left(3\cdot[6^*7_{10}]_{n-1}\right)\left([49^*_{10}]_n\right)/3\] \[[6^*56^*_{10}]_n={20\cdot10^{2n}-3\cdot10^{n}-2\over3}=\left(4\cdot10^{n}+1\right)\left(5\cdot10^{n}-2\right)/3=\left([40^*1_{10}]_{n-1}\right)\left(6\cdot[83^*_{10}]_{n-1}\right)/3\] \[[6^*76^*_{10}]_n={20\cdot10^{2n}+3\cdot10^{n}-2\over3}=\left(4\cdot10^{n}-1\right)\left(5\cdot10^{n}+2\right)/3=\left(3\cdot[13^*_{10}]_n\right)\left([10^*4_{10}]_n/2\right)/3\]