Sequence relating to Pell numbers; a new technique

[Mitchell Harris]

On Wed, 6 Oct 2004, Mitchell Harris wrote:

>On Sun, 3 Oct 2004 creigh at o2online.de wrote:
>
>>Surely this relation is already known...?
>
>Any 2nd order linear recurrence with constant coefficients is a linear 
>combination of successive Fibonacci #'s:
>
>Let G(n) = a G(n-1) + b G(n-2), with G(0) and G(1) the given base cases 
>(i.e. constants but... er... variable).
>
>Then G(n) happens to be 
>  G(1) F(n) + G(0) F(n-1) (proof by induction)

Er...no, that's pretty wrong. It is certainly true when a = b = 1.

Pardon my repeated mail errors (this last one was a draft that I was 
supposedly trying to delete but I sent it instead).

>so in one sense, all this playing around with such recurrences is like 
>playing around with addition: given two random 10 digits #'s, it is 
>unlikely that their addition has ever been computed before.
>
>so
>1) this relation may not have ever been written down before (if that's 
>what you mean by "is already known") (so it's new)
>2) but that's not surprising, because any particular identity wil not be 
>terribly earth shattering. (so it's not really new)
>3) but it's how you're using these things (these new but not new 
>things) that is/can be interesting.

So my point (which I was trying not to make by not sending but am now 
forced to because of my mistake), is that 1) there are always intersting 
identities to be found and similarly 2) there are always new sequences 
which aren't yet in the OEIS.

Mitch