The Kimberling Sequence
Julia, 78 71 bytes



n-&gt;[(i=j=x;while j&lt;2i-3 j=i-(j%2&gt;0?1-j:j+2)÷2;i-=1end;i+j-1)for x=1:n]


This is an unnamed function that accepts an integer and returns an integer array. To call it, assign it to a variable.

The approach here is the same as that described on OEIS.

Ungolfed:

# This computes the element of the sequence
function K(x)
    i = j = x
    while j &lt; 2i - 3
        j = i - (j % 2 &gt; 0 ? 1 - j : j + 2)÷2
        i -= 1
    end
    return i + j - 1
end

# This gives the first n terms of the sequence
n -&gt; [K(i) for i = 1:n]


Saved 7 bytes thanks to Mauris!