Neither 1 nor 36 can be created by adding any two numbers in the range 1 - 17 together.

First I looked at the neighbours each number could have

1 neighbour
-----------
16 -> 9
17 -> 8

16 and 17 can only have one neigbour therefore they must be at each end of the sequence.

2 neighbours
------------
2 -> 14, 7
4 -> 12, 5
5 -> 4, 11
6 -> 10, 3
7 -> 2, 9
8 -> 17, 1
9 -> 7, 16
10 -> 15, 6
11 -> 5, 14
12 -> 13, 4
13 -> 3, 12
14 -> 11, 2
15 -> 1, 10

3 neighbours
------------
1 -> 8, 15, 3
3 -> 6, 13, 1

As stated earlier the sequence must start with either 16 or 17. If we start with 17, it must be followed 
by 8 because it only has only one neighbour. Then 8 must be followed by 1 (we've already used 17) and this
process must carry on until we reach a number with 3 neighbours. The diagram below shows this:

                --13, 12, 4, 5, 11, 14, 2, 7, 9, 16 (this path misses numbers so can be ruled out)
                |
           -- 3-
          |     |
17, 8, 1 --     --6, 10, 15, (This can be ruled out as we would need to reuse 1) 
          |
          |             --(Even though 3 has 3 neighbours we can only use the number 13 as others have been used) 
          |             |
          --15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16           
          
          
The diagram above shows that the only order of numbers that works is:

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

Of course, this could be reversed to start with 16.