Solution Re: Request for information about basis representation

[Ralf Stephan]

You wrote 
> The row T(a, n) for fixed a is an a-regular set, that is, its a-ary 
> representations
> form an regular language over the alphabet of a-ary digits, namely (0+1)*.
> This will be 2-regular only for a = 2^k, k >= 1. 

No. According to Allouche/Shallit[1], what you define is 2-automaticity.

That T(3,n) is 2-regular is also explicitly stated in [1] where 
Example 3, p.25 is just 2*T(3,n) shifted.


Regards,
R Stephan

[1] J.-P. Allouche and J. Shallit, The ring of k-regular sequences, Theoretical Computer Sci., 98 (1992), 163-197.
also in:
[2] J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge Univeristy Press, 2003.