Up/Down-Related Sequence(s)
Leroy Quet
qq-quet at mindspring.com
Mon Dec 15 01:54:39 CET 2003
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
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