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

[Joshua Zucker]

On 2/1/06, Max <relf at unn.ac.ru> wrote:
> Paul D. Hanna wrote:
> > (A1) "a(n) = number of unique matrix products in (A+B+C+D)^n  where
> > commutator [A,B]=0 but neither A nor B commutes with C or D."
>
> It is A001353
>
> > (A2) "a(n) = number of unique matrix products in (A+B+C+D)^n  where
> > commutators [A,B]=[C,D]=0 but neither A nor B commutes with C or D."
>
> a(n) = 2 * A003480(n)

I think it's actually A007070 -- assuming a(0) = 1 is correct.  That's
the only difference between A007070 and 2*A003480, anyway.

>
> > (A3) "a(n) = number of unique matrix products in (A+B+C+D)^n  where
> > commutators [A,B]=[A,C]=[B,C]=0 but D does not commute with A, B, or C."
>
> It is A052529

I think there's an offset of 1:  It is A052529(n+1).

I wrote an excruciatingly slow program to enumerate all these
different strings of ABCDs, and I confirm all of the above results at
least up through the first few terms.

Again, Paul and/or Max, I think these would make great comments for
the given sequences!

Thanks for the fun problem.

--Joshua Zucker