Base 4
Patterns that can be factored: \[[1^*01^*_4]_n={4\cdot4^{2n}-3\cdot4^{n}-1\over3}=\left(4^{n}-1\right)\left(4\cdot4^{n}+1\right)/3=\left(3\cdot[1^*_4]_n\right)\left([10^*1_4]_n\right)/3\] \[[1^*21^*_4]_n={4\cdot4^{2n}+3\cdot4^{n}-1\over3}=\left(4^{n}+1\right)\left(4\cdot4^{n}-1\right)/3=\left([10^*1_4]_{n-1}\right)\left(3\cdot[1^*_4]_{n+1}\right)/3\] \[[2^*02^*_4]_n=2\cdot{4\cdot4^{2n}-3\cdot4^{n}-1\over3}=2\cdot\left(4^{n}-1\right)\left(4\cdot4^{n}+1\right)/3=2\cdot\left(3\cdot[1^*_4]_n\right)\left([10^*1_4]_n\right)/3\] \[[3^*03^*_4]_n=4\cdot4^{2n}-3\cdot4^{n}-1=\left(4^{n}-1\right)\left(4\cdot4^{n}+1\right)=\left(3\cdot[1^*_4]_n\right)\left([10^*1_4]_n\right)\]
Links
> 10001: Depression 100..001
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> 10003: Quasi Repdigit 100..003
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> 10111: Near Repdigit 1011..11
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> 11101: Near Repdigit 11..1101
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> 11111: Repunit 11..11
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> 11113: Near Repdigit 11..113
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> 11121: Near Repdigit 11..1121
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> 11131: Near Repdigit 11..1131
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> 11311: Near Repdigit Palindrome 11..11311..11
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> 12111: Near Repdigit 1211..11
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> 12223: Quasi Repdigit 122..223
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> 13111: Near Repdigit 1311..11
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> 13331: Plateau 133..331
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> 13333: Near Repdigit 133..33
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> 20003: Quasi Repdigit 200..003
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> 21111: Near Repdigit 211..11
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> 21112: Depression 211..112
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> 21113: Quasi Repdigit 211..113
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> 21222: Near Repdigit 2122..22
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> 22122: Near Repdigit Palindrome 22..22122..22
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> 22212: Near Repdigit 22..2212
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> 22221: Near Repdigit 22..221
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> 22223: Near Repdigit 22..223
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> 22322: Near Repdigit Palindrome 22..22322..22
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> 23222: Near Repdigit 2322..22
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> 23332: Plateau 233..332
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> 23333: Near Repdigit 233..33
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> 30001: Quasi Repdigit 300..001
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> 30002: Quasi Repdigit 300..002
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> 31111: Near Repdigit 311..11
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> 31333: Near Repdigit 3133..33
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> 32221: Quasi Repdigit 322..221
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> 32223: Depression 322..223
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> 32333: Near Repdigit 3233..33
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> 33133: Near Repdigit Palindrome 33..33133..33
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> 33233: Near Repdigit Palindrome 33..33233..33
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> 33313: Near Repdigit 33..3313
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> 33323: Near Repdigit 33..3323
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> 33331: Near Repdigit 33..331
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