a(n)=Number of Unique Matrix Products in (A+B+C)^n When [A,B]=0

[Paul D. Hanna]

Seqfans, 
      What is the sequence defined as follows: 
  
a(n) = the number of unique products of matrices A, B, C, in (A+B+C)^n 
where commutator [A,B]= 0 but C does not commute with A or B. 
  
The sequence begins:  {1, 3, 8, 21, ...}  since 
n=0 has 1 unique product: 
(A+B+C)^0 = I 
n=1 has 3 unique products:
(A+B+C)^1 = A+B+C 
n=2 has 8 unique products:
(A+B+C)^2 = AA + BB + CC + 2AB + AC + CA + BC + CB   
n=3 has 21 unique products:
(A+B+C)^3 = AAA + 3AAB + 3ABB + BBB + AAC + ACA + CAA +
BBC + BCB + CBB + 2ABC + ACB + BCA + 2CAB +
ACC + CAC + CCA + BCC + CBC + CCB + CCC
...
Most likely this is already in the OEIS ... 
 
Variations of this would include: 
"a(n) = number of unique matrix products in (A+B+C+D)^n  where 
commutator [A,B]=[A,D]=[B,D]=0 but D does not commute with A, B, or C." 
 
and other obvious variations could be explored. 
      Paul