Constant C=0.1688... for A081881 seems to be wrong

[Max Alekseyev]

Rainer,

If I understand correctly, David's formula is just a heuristics that
has not been proved to be correct for all n yet. If so, it should be
used carefully and not stated as an established fact in OEIS.
If not and I missed the proof of David's formula, I would be very
grateful to see it.

Regards,
Max

On Jan 23, 2008 3:34 PM, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> Thanks to Neil's latest update I am proud to
> announce A136616 and A136617, dealing with
> harmonic numbers.
>
> #
> # 1. Extending A081881
> #
>
> A related sequence is Wouter Meeussen's A081881.
> Using the relation with A136617 and the nice
> formula from David Cantrell there, it is easy to
> extend this sequence from 12 elements now to many
> more. The first 40 elements are:
>
> 1, 2, 4, 10, 26, 69, 186, 504, 1369, 3720, 10111,
> 27483, 74705, 203068, 551995, 1500477, 4078718,
> 11087104, 30137872, 81923228, 222690421, 605335323,
> 1645472007, 4472856655, 12158484965, 33050188741,
> 89839727480, 244209698681, 663830786257, 1804479163453,
> 4905082919846, 13333397768101, 36243932864644,
> 98521224097850, 267808453182726, 727978851794328,
> 1978851684335001, 5379076574743407, 14621846107014725,
> 39746298571222758, 108041641154662534
>
> The Maple code is:
> restart:Digits:=50:e:=exp(1);A136617 := n ->
> floor( (e - 1)*(n - 1/2) + (e - 1/e)/(24*(n - 1/2)) );
> apq := n -> n + A136617(n);A081881 := n -> (apq@@n)(1);
> seq(A081881(n),n=0..40);
>
> #
> # 2. Checking Benoit Cloitre's formula
> #
>
> The FORMULA section in A081881 says:
> a(n) is asymptotic to C*exp(n) where C=0.1688...
> - Benoit Cloitre (abmt(AT)wanadoo.fr), Apr 14 2003
>
> Numerical evidence suggests that this limit exists:
>
>                         a(n)
>          C  =   lim   -------
>                n->oo      n
>                          e
>
> but the value is  C = 0.4589991659480974... as it seems
> and not 0.1688...
>
> I would like to add my Maple code to A081881 and give a
> cross reference to A136617 and David Cantrell's formula.
> At the same time I would like to correct the formula of
> Benoit Cloitre. But I may be wrong and so I ask you as
> SeqFan.
>
> Best regards,
> Rainer Rosenthal
> r.rosenthal at web.de
>
>
>