[seqfan] Re: Hofstadter's A005228, differences=complement

[Neil Sloane]

Benoit,

That is really excellent work - thanks!

Could you add this information to A005228 and A030124?

It could take the form of a plain text file or a pdf file attached
to those entries.

Neil


On Tue, May 21, 2013 at 5:26 PM, Benoît Jubin <benoit.jubin at gmail.com>wrote:

> I obtained an asymptotic expansion for these sequences to arbitrary
> order: let u(n) = A030124(n) - n. Then
> u(n) ~ \sum_{k>=1} c(k) (n/2)^{1/2^k}
> where
> c(k) = (-1)^{k+1} 2^Fibo(k+1) / (prod_{m=1}^{k-1} (2^m+1)).
> One then finds the asymptotic expansion of A005228 by integration.
>
> I used the relation
> A005228(u_n) = n + O(u_n),
> which follows from the partition condition, to obtain via integration
> u_n^2 + 2 \sum_{k=1}^{u_n} u_k = 2n + O(\sqrt{n}).
> From this formula, one obtains the above expansion by induction.
>
> Benoît Jubin
>
>
> On Mon, May 20, 2013 at 4:35 AM, Neil Sloane <njasloane at gmail.com> wrote:
> > I also thought the offset was strange, and since
> > you said the same thing, I just changed it!
> >
> >
> > On Sun, May 19, 2013 at 10:15 PM, Benoît Jubin <benoit.jubin at gmail.com
> >wrote:
> >
> >> > Right now I don't even know a proof that A030124(n) is asymptotic to
> n,
> >> although this should not be too difficult.
> >>
> >> This has probably been noticed by the interested parties already, but
> >> here it is:
> >> Since the sequence of first differences of A005228 is strictly
> >> increasing, one has A005228(n) >= n(n+1)/2. Therefore A030124(n-1) is
> >> bounded by the complement of the sequence (n(n+1)/2), call it b.
> >> If (n-1)n/2 < k <= n(n+1)/2, then b(k) = (n+1)(n+2)/2 - 1 - ( n(n+1)/2
> >> - k) = n + k, so computing n in terms of k, one has
> >> A030124(k-1) <= k + ceil( sqrt(2k+1/4) - 1/2)
> >> and for k large enough (for instance k>100), one has
> >> k < A030124(k-1) < k + sqrt(2k)
> >>
> >> (the offset of A030124 is peculiar)
> >>
> >> Benoît
> >>
> >>
> >> On Sun, May 19, 2013 at 5:17 PM, Neil Sloane <njasloane at gmail.com>
> wrote:
> >> > Alan, thanks for telling me. In fact both b-files (A005228 and
> A030124)
> >> > were wrong, and I have now corrected them.
> >> >
> >> > Right now I don't even know a proof that A030124(n) is
> >> > asymptotic to n, although this should not be too difficult.
> >> > I still don't have any rigorous bounds for either sequence.
> >> >
> >> > Neil
> >> >
> >> >
> >> > On Sun, May 19, 2013 at 9:33 AM, Allan Wechsler <acwacw at gmail.com>
> >> wrote:
> >> >
> >> >> I don't have an answer to the question, but the graph at A005228
> looks
> >> >> wrong. It looks like the graph for A030124 instead.
> >> >>
> >> >>
> >> >> On Wed, May 15, 2013 at 7:25 PM, Neil Sloane <njasloane at gmail.com>
> >> wrote:
> >> >>
> >> >> > Has anyone seen any rigorous bounds on A005228(n)
> >> >> > or A030124(n)?
> >> >> > Neil
> >> >> >
> >> >> > _______________________________________________
> >> >> >
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> >> >> >
> >> >>
> >> >> _______________________________________________
> >> >>
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> >> >>
> >> >
> >> >
> >> >
> >> > --
> >> > Dear Friends, I have now retired from AT&T. New coordinates:
> >> >
> >> > Neil J. A. Sloane, President, OEIS Foundation
> >> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> >> > Phone: 732 828 6098; home page: http://NeilSloane.com
> >> > Email: njasloane at gmail.com
> >> >
> >> > _______________________________________________
> >> >
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> >>
> >> _______________________________________________
> >>
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> >>
> >
> >
> >
> > --
> > Dear Friends, I have now retired from AT&T. New coordinates:
> >
> > Neil J. A. Sloane, President, OEIS Foundation
> > 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
> > Phone: 732 828 6098; home page: http://NeilSloane.com
> > Email: njasloane at gmail.com
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>



-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com