When playing around with base 7, I didn't expect to find any interesting patterns, but:

2x table: 2, 4, 6, 11, 13, 15, 20, 22, 24, 26, 31, 33,

*... all the numbers whose digit sum is even!*

3x table: 3, 6, 12, 15, 21, 24, 30, 33, 36,

*... all the numbers in the 3x table in base-10 that are possible in base-7!*

4x table: 4, 11, 15, 22, 26, 33, 40, 44, 51, 55,

*... every second number is in the 11 times table in base 10!*

5x table: 5, 13, 21, 26, 34, 42, 50, 55, 63, 71, 76, 84, 92, 100, ...

*mmm?*

6x table: 6, 15, 24, 33, 42, 51, 60, 66, 75, 84, 93, 102, 110, ...

*like the 9x table in base 10 (not that surprising!).*

Also, the square numbers are interesting:

1, 4, 12, 24, 34, 51, 100, 121, 144, 202, 232, 264, 331, 400, 441, ...

Well, maybe not that interesting until you compare them to square numbers in other bases:

Base-3: 1, 11, 100, 121, 221, ...

Base-4: 1, 10, 21, 100, 121, 210, ...

Base-5: 1, 4, 14, 31, 100, 121, 144, ...

Base-6: 1, 4, 13, 24, 41, 100, 121, 144, ...

Base-7: 1, 4, 12, 24, 34, 51, 100, 121, 144, ...

Base-8: 1, 4, 11, 20, 31, 44, 61, 100, 121, 144, ...

Base-9: 1, 4, 10, 17, 27, 40, 54, 71, 100, 121, 144, ...

Base-10: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ...

Base-11: 1, 4, 9, 15, 23, 33, 45, 59, 74, 91, 100, 121, 144, ...

Base-12: 1, 4, 9, 14, 21, 30, 41, 54, 69, 84, A1, 100, 121, 144, ...