a[n] is a[n] terms from term = a[n]+1

[Leroy Quet]

You have a sequence where:
a[1] = 1;
and either a[n -a[n]] = a[n]+1
or
a[n +a[n]] = a[n]+1

such that a[n -a[n]] is chosen whenever it is not equal
to any positive integer <= a[n] already.

The sequnce begins, with 0 representing "empty" a[n]'s:

1, 2, 5, 3, 0, 0, 4, 6, 17, 15, 13, 11, 9, 7, 0, 
0, 0, 0, 0, 0, 8, 10, 12, 14, 16, 18,...

(If I didn't err...)

Does this sequence continue forever (so that no a[n] = more than one
different value at once, of course)?

And if you are free to choose

a[n -a[n]] = a[n]+1
or
a[n +a[n]] = a[n]+1,

then is it possible to get a sequence without any zeros at all?

Thanks,
Leroy Quet