One more contribution.
Let as in the previous post \( w(x) \) be the generating function for the original sequence A \( (1,1,2,2,4,4,6,6,10,10,...) \) .
Now we look at the sequences \( A_k \) for which the powers \( w(x)^k \) are the generating functions. In the previous posts I'd already pointed to that of k=-1 where the sequence \( A_{-1} \) is the "ubiquitueous Thue-Morse-sequence".
First a list of the sequences. Left column is for \( w(x)^{-3} \) , 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 \( M_k \) and the associated sequence \( A_k \) 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 \( w(x) \) be the generating function for the original sequence A \( (1,1,2,2,4,4,6,6,10,10,...) \) .
Now we look at the sequences \( A_k \) for which the powers \( w(x)^k \) are the generating functions. In the previous posts I'd already pointed to that of k=-1 where the sequence \( A_{-1} \) is the "ubiquitueous Thue-Morse-sequence".
First a list of the sequences. Left column is for \( w(x)^{-3} \) , 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 \( M_k \) and the associated sequence \( A_k \) 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