Comments from _Paul Curtz_, Jan 25 2016:
The positions of the 1's are given by A003622.

n = 1,2,3,4,5,... appears in position 1,2,3,5,8, ...  = A000045(n+2).
The sequence may be written as follows:
1, 2, 3,
1, 4,
1, 2, 5,
1, 2, 3,
1, 6,
1, 2, 3,
1, 4,
1, 2, 7,
etc.
The rows have c(n) = 3, 2, 3, 3, 2, 3, 2, 3, 3, 2, 3, 3, ... = A003622(n+2) -A003622(n+1) terms
Second column (evens) : A255671(n+1).
a(n+1) - a(n) =
1,  1, -2,
3, -3,
1,  3, -4,
1,  1, -2,
5, -5,
1,  1, -2,
3, -3,
1,  5, -6,
etc.
Row sums = 0.
Removing all 1's from a(n) gives
2
3
4     2,
5     2, 3,
6     2, 3, 4, 2,
7     2, 3, 4, 2, 5, 2, 3,
8     2, 3, 4, 2, 5, 2, 3, 6, 2, 3, 4, 2,
etc.
Row sums: A001610(n+1). Compact terms row sums: A023548(n).
(See A268034.) 

a(n) is the second sequence of the array
0, 1, 2, 0, 3, 0, 1, 4, 0, ...   A035614(n)
1, 2, 3, 1, 4, 1, 2, 5, 1, ...
2, 3, 4, 2, 5, 2, 3, 6, 2, ...   a2(n)
3, 4, 5, 3, 6, 3, 4, 7, 3, ...   a3(n)
4, 5, 6, 4, 7, 4, 5, 8, 4, ...
etc.
a(n) is A035614(n) without the 0's. Also A035614(n) + 1.
a2(n) is a(n) without the 1's. Also a(n) + 1.
etc.
From a(n) to A268034(n+1).
a(n) + 2 = a3(n).
A001045(a3(n)) = 3, 5, 11, 3, 21, ... = A268034(n+1).