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

[Benoît Jubin]

> 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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> --
> 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
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