Up/Down-Related Sequence(s)

[Leroy Quet]

>I had this idea for a recursively-defined zig-zagging (though not 
>entirely strictly so) sequence (not in the EIS, I believe):
>
>
>a(1)  = 1;
>
>A(1) = = [a(1)] = [1].
>
>A(m+1) = [A(m),A(m)+(-1)^a(m)],
>
>where A(m+1) = a finite sequence (of 2^m terms) which is A(m) followed by 
>each shifted term of A(m),
>where each term in the second A(m) is one higher or one lower than in the 
>previous 
>depending upon the m_th individual term a(m) of the total sequence.
>
>
>
>Sequence:
>
>A(6) ->
>
>1,  0,  2, 1,  2, 1, 3, 2,  0, -1, 1, 0, 1, 0, 2, 1, 
>2, 1, 3, 2, 3, 2, 4, 3, 1, 0 , 2, 1, 2, 1, 3, 2
>
>
>.....


Ah, this seems to be (1 - (A010060))(mod 2), where A010060 is the 
Thue-Morse sequence.

(And the a(1)=0 sequence is simply congruent to A010060 mod2.)


So, A(m+1) would  = [A(m), A(m) +(-1)^(a(1)+(A010060))].

thanks,
Leroy Quet