11111: Repunit 11..11

First few terms (base 10): \[[1^*_9]_n=\{0,1,10,91,820,\ldots\}\]

First few terms (base 9): \[\{\text{},\text{1},\text{11},\text{111},\text{1111},\ldots\}\]

General Formula:\[[1^*_9]_n={9^{n}-1\over8}\]

Equivalent Patterns: \[[2^*_9]_n=2\cdot[1^*_9]_n\] \[[4^*_9]_n=4\cdot[1^*_9]_n\] \[[8^*_9]_n=8\cdot[1^*_9]_n\] \[[18^*7_9]_n=16\cdot[1^*_9]_{n+1}\] \[[38^*5_9]_n=32\cdot[1^*_9]_{n+1}\] \[[5^*_9]_n=5\cdot[1^*_9]_n\] \[[12^*1_9]_n=10\cdot[1^*_9]_{n+1}\] \[[24^*2_9]_n=20\cdot[1^*_9]_{n+1}\] \[[48^*4_9]_n=40\cdot[1^*_9]_{n+1}\] \[[7^*_9]_n=7\cdot[1^*_9]_n\] \[[16^*5_9]_n=14\cdot[1^*_9]_{n+1}\] \[[34^*1_9]_n=28\cdot[1^*_9]_{n+1}\] \[[68^*2_9]_n=56\cdot[1^*_9]_{n+1}\] \[[78^*1_9]_n=64\cdot[1^*_9]_{n+1}\] \[[13^*2_9]_n=11\cdot[1^*_9]_{n+1}\] \[[26^*4_9]_n=22\cdot[1^*_9]_{n+1}\] \[[15^*4_9]_n=13\cdot[1^*_9]_{n+1}\] \[[23^*1_9]_n=19\cdot[1^*_9]_{n+1}\] \[[46^*2_9]_n=38\cdot[1^*_9]_{n+1}\] \[[27^*5_9]_n=23\cdot[1^*_9]_{n+1}\] \[[56^*1_9]_n=46\cdot[1^*_9]_{n+1}\] \[[35^*2_9]_n=29\cdot[1^*_9]_{n+1}\] \[[37^*4_9]_n=31\cdot[1^*_9]_{n+1}\] \[[45^*1_9]_n=37\cdot[1^*_9]_{n+1}\] \[[57^*2_9]_n=47\cdot[1^*_9]_{n+1}\] \[[67^*1_9]_n=55\cdot[1^*_9]_{n+1}\] \[[3^*_9]_n=3\cdot[1^*_9]_n\] \[[6^*_9]_n=6\cdot[1^*_9]_n\] \[[14^*3_9]_n=12\cdot[1^*_9]_{n+1}\] \[[28^*6_9]_n=24\cdot[1^*_9]_{n+1}\] \[[58^*3_9]_n=48\cdot[1^*_9]_{n+1}\] \[[17^*6_9]_n=15\cdot[1^*_9]_{n+1}\] \[[36^*3_9]_n=30\cdot[1^*_9]_{n+1}\] \[[25^*3_9]_n=21\cdot[1^*_9]_{n+1}\] \[[47^*3_9]_n=39\cdot[1^*_9]_{n+1}\]

Table

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