Straight Chain Alk*nes without Order This is much like my earlier challenge, except, this time, order doesn't matter.

A straight-chain alk*ne is defined as a sequence of carbon atoms connected by single (alkane), double (alkene), or triple bonds (alkyne), (implicit hydrogens are used.) Carbon atoms can only form 4 bonds, so no carbon atom may be forced to have more than four bonds. A straight-chain alk*ne can be represented as a list of its carbon-carbon bonds.

These are some examples of valid (not necessarily distinct) straight-chain alk*nes:

[]       CH4              Methane
[1]      CH3-CH3          Ethane
[2]      CH2=CH2          Ethene
[3]      CH≡CH            Ethyne
[1,1]    CH3-CH2-CH3      Propane
[1,2]    CH3-CH=CH2       Propene
[1,3]    CH3-C≡CH         Propyne
[2,1]    CH2=CH-CH3       Propene
[2,2]    CH2=C=CH2        Allene (Propadiene)
[3,1]    CH≡C-CH3         Propyne 
[1,1,1]  CH3-CH2-CH2-CH3  Butane
...


While these are not, as at least one carbon atom would have more than 4 bonds:

[2,3]
[3,2]
[3,3]
...


Two straight-chain alk*nes, p and q are considered equivalent if p is q reversed, or p is q.

[1] = [1]
[1,2] = [2,1]
[1,3] = [3,1]
[1,1,2] = [2,1,1]
[1,2,2] = [2,2,1]


Your task is to create a program/function that, given a positive integer n, outputs/returns the number of valid straight-chain alk*nes of exactly n carbon atoms in length.

Specifications/Clarifications


You must handle 1 correctly by returning 1.
Alk*nes like [1,2] and [2,1] are NOT considered distinct.
Output is the length of a list of all the possible alk*nes of a given length.
You do not have to handle 0 correctly.


Test Cases:

1 =&gt; 1
2 =&gt; 3
3 =&gt; 4
4 =&gt; 10
5 =&gt; 18
6 =&gt; 42


This is code golf, so the lowest byte count wins!
JavaScript (ES6), 107 bytes

A=n=&gt;n&gt;9?2*A(n-1)+3*A(n-2)+3*A(n-7)+A(n-8)-5*A(n-3)-A(n-4)-2*A(n-6)-A(n-9):[0,1,3,4,10,18,42,84,192,409][n]


Recursive solution, using the recurrence relation that was pointed out in the comments of the question. Execution time rises much quicker than the input (complexity of O(9^N) if I'm not mistaken), so be careful with values higher than 20.