[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
your n x a,
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])[1] = n
and (n x [1,1])[2] = 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[1]+b[1] , a[1]*b[2]+a[2]*b[1]
>
> 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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