[seqfan] Re: Near-linear sequence

[Neil Sloane]

Concerning A101402, the terms fall naturally into blocks of sizes
1,1,1,2,4,8,16,32,...:

0,

1,

1,

1, 2,

2, 3, 3, 3,

3, 4, 4, 4, 5, 5, 6, 6,

6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12,

12, 13, 13, 13, 14, 14, ...

Then the definition says that the k-th block is the final term of the
previous block added to the sequence starting from the beginning.
(Eg 34445566 = 3+01112233)

The final terms of the blocks, a(2^k), appear to be given by A164363.
Is that obvious?

Neil

On Wed, Aug 27, 2014 at 12:39 PM, Charles Greathouse
<charles.greathouse at case.edu> wrote:
> Sequence A101402 appears to be nearly linear. For the first 10,000 terms
> there is a constant k such that |a(n) - kn| < 2 (e.g., take k = 0.355). Can
> anyone prove or disprove that a(n) = kn + O(1) for some constant k? In the
> (likely?) latter case, can another reasonable bound be found, maybe O(log
> n)? I can't even think of a technique that would work here.
>
> I just checked to a million and it looks like the same holds. Here I used k
> = 0.3549419505. Probably going to 10 million would require relaxing the
> bound slightly; already by a million the choice of constant is very
> constrained.
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
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>
> 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.
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