[seqfan] Re: [ Re: A080605 Golomb seq samples extra 23]
Benoit CLOITRE
benoit7848c at orange.fr
Sat Jun 16 12:05:17 CEST 2012
"Likewise A080606 evens and A080607 triples?"
Yes. I will make the changes to corresponding entries. In general if you define a Golomb variation a(n) using a(1)=x+y and the arithmetical progression x*n+y instead of n (x>=1, y>=0) you have the formula:
(i) a(a(1)+a(2)+...+a(n))=x*n+y
The case A080605 yields a(a(1)+..a(n))=2n-1 and it is easy to see where the asymptotic formula comes from, Indeed suppose a(n)\sim \alpha n^\beta then you have to solve a simple equation using (i) to got alpha and beta values. Vardi paper explains how to make this rigourous.
There are many Golomb variations but we could define a Golomb's transform of any strictly increasing sequence b(n) as follows for n>=1:
The Golomb transform of b is the earliest monotonic sequence satisfying a(1)=b(1) and a(a(1)+..+a(n))=b(n).
For instance the Golomb transform of integers is A001462, the Golomb transform of odd integers is A080605 and the Golomb transform of primes would be A169682 with pari code:
a=[2,2];for(n=2,50,for(i=1,a[n],a=concat(a,prime(n))));a
Some work is needed to got the asymptotic formula for a(n) in this case.
> Message du 16/06/12 09:50
> de : "KevinRyde"<user42 at zip.com.au>
> à : seqfan at list.seqfan.eu
> cc :
> objet : [seqfan] Re: A080605 Golomb seq samples extra 23
>
> benoit7848c at orange.fr (Benoit Cloitre) writes:
> >
> > Also my comment A080605(n)=tau^(2-tau)*(2n)^(tau-1)+O(1) is wrong
> > since Vardy method gives O(n^(tau-1)/log(n)) instead of O(1).
>
> Ah yes. Likewise A080606 evens and A080607 triples?
> One of the links related to that in A001462 may have moved,
>
> Y.-F. S. Petermann, J.-L. Remy and I. Vardi, Discrete derivatives of
> sequences, Adv. in Appl. Math. 27 (2001), 562-84.
>
> to
>
> http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/publications.html
> http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/discrete_derivatives.ps
>
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