New Neighbour Sequence
JavaScript (ES6), 13 bytes
Returns the \$n\$th term of the sequence.
n=&gt;n-2-~++n%5

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How?
This computes:
$$n-2+((n+2) \bmod 5)$$
           n |  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 ...
-------------+--------------------------------------------------
       n - 2 | -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 ...
 (n+2) mod 5 |  2  3  4  0  1  2  3  4  0  1  2  3  4  0  1 ...
-------------+--------------------------------------------------
         sum |  0  2  4  1  3  5  7  9  6  8 10 12 14 11 13 ...