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


Or, if we let a(1) = 0, A(5):

0,  1,  -1, 0,  -1, 0, -2, -1,  1, 2, 0, 1, 0, 1, -1, 0


So, I guess, if a(n,m) is the n_th term of the sequence with a(1) = m,

then a(n,m) = m +(-1)^m *a(n,0).


What can be said about this sequence, such as the expected upper/lower 
bounds, the number of 0's before a certain term, etc?

 
And we can create related sequences, such as:

A(m+1) = [A(m),A(m)+ b(m)(-1)^a(m)],

where b(m), the amount added/subtracted on each iteration, is a term of 
any integer sequence (including possibly of this sequence itself).


thanks,  
Leroy Quet