[seqfan] Re: Hofstadter's A005228, differences=complement
Neil Sloane
njasloane at gmail.com
Mon May 20 04:35:41 CEST 2013
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
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