# [seqfan] Re: sequence defined by ``exotic addition''

Maximilian Hasler maximilian.hasler at gmail.com
Tue Dec 22 17:04:13 CET 2009

```some remarks :

1/
Note that the 1st component of n x [1,1] is given in A087808.

2/
I don't think the terminology "scalar multiplication" is adequate for
since it does not very any property usually attributed to scalar multiplication,
like e.g.  (nm) x a = n x (m x a)
and/or  (n+m) x a = (n x a) o (m x a)
and/or  n x (a o b) = n x a o n x b.

(I think none of these necessarily implies associativity of "o".)

It's simply a recurrent definition of a (2-component vector) sequence.

3/
You could also define
n x a  = ((n-1) x a ) o a
In this case (n x [1,1]) = n
and  (n x [1,1]) = A000124(n)

Maximilian

On Tue, Dec 22, 2009 at 10:28 AM, Georgi Guninski <guninski at guninski.com> wrote:
> define binary operation "o" on pairs a,b:
> a o b = a+b , a*b+a*b
>
> define scalar multiplication "x":
> 2n x a = (n x a) o (n x a)
> 2n+1 x a= ((n x a) o (n x a)) o a
> 1 x a = a
>
>
> the sequence a(n) is the second component of n x [1,1]:
> 1,2,4,8,12,24,40,64,72,144,224,320,416,576,768,1024,1040 ...
>
> the sequence may be more interesting if "o" were associative, is there a
> list of binary associative functions?
>
> attached is a sample pari implementation.
>
> superseeker claims to find degree 15 revogf.
>
> thanks.
>
> --
> georgi
>
>
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>
>

```