One more contribution.

Let as in the previous post be the generating function for the original sequence A .

Now we look at the sequences for which the powers are the generating functions. In the previous posts I'd already pointed to that of k=-1 where the sequence is the "ubiquitueous Thue-Morse-sequence".

First a list of the sequences. Left column is for , then for k=-2,-1,0,1 (which gives the column with our sequence under discussion), then k=2,3,4:

Now the matrices, for which that sequences are the "eigen-sequences".

Their generating follows a very simple pattern.

I always show the matrix and the associated sequence together:

(for sequence and more information see http://oeis.org/A106407 )

(for sequence and more information see http://oeis.org/A106400 )

(for sequence and more information see http://oeis.org/A018819 )

Let as in the previous post be the generating function for the original sequence A .

Now we look at the sequences for which the powers are the generating functions. In the previous posts I'd already pointed to that of k=-1 where the sequence is the "ubiquitueous Thue-Morse-sequence".

First a list of the sequences. Left column is for , then for k=-2,-1,0,1 (which gives the column with our sequence under discussion), then k=2,3,4:

Now the matrices, for which that sequences are the "eigen-sequences".

Their generating follows a very simple pattern.

I always show the matrix and the associated sequence together:

(for sequence and more information see http://oeis.org/A106407 )

(for sequence and more information see http://oeis.org/A106400 )

(for sequence and more information see http://oeis.org/A018819 )

Gottfried Helms, Kassel