Solution Re: Request for information about basis representation

[David Wilson]

----- Original Message ----- 
From: "Ralf Stephan" <ralf at ark.in-berlin.de>
To: <ham>; "Andrew Plewe" <aplewe at sbcglobal.net>
Cc: "'Sequence Fans'" <seqfan at ext.jussieu.fr>
Sent: Saturday, March 05, 2005 10:48 AM
Subject: Solution Re: Request for information about basis representation


> Let T(a,n) be the number 'replace 2^i with a^i in binary representation of 
> n'.
> The columns (n fixed) are all polynomials in a. The rows are 2-regular
> sequences. Most 2-regular sequences can be expressed in terms of A007814,
> the dyadic valuation of n (from that, they have all the same kind of
> fractality). The sequences at hand are no exception:

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.