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

[Max]

Paul D. Hanna wrote:
> 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 

This sequence satisfies the recurrence: a(n)=3*a(n-1)-a(n-2)
and can be explicitly expressed as
a(n) = (5+3*sqrt(5))/10*((3+sqrt(5))/2)^n + (5-3*sqrt(5))/10*((3-*sqrt(5))/2)^n

It is in the OEIS under the name A001906

Max