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

[Peter Pein]

Hello Rainer,

you've produced a wrong offset with your Maple code.

It should read:
A081881 := n -> (apq@@(n-1))(1);
                       ^^^
seq(A081881(n),n=1..40);
                ^^^

Using the offset 1 (as in OEIS), I get with Mathematica:

In[1]:=
a136617[n_] := Block[{$MaxExtraPrecision = 1000},
    Floor[(E - 1)*(n - 1/2) + (E - 1/E)/(24*(n - 1/2))]];
value256 = Nest[a136617[#1] + #1 & , 1, 255];
In[3]:=
BenoitConst = N[value256/Exp[256], 30]
Out[3]=
0.168856356667144203731679775501
In[4]:=
RainerConst = N[value256/Exp[255], 30]
Out[4]=
0.458999165948097439647704779206
In[5]:=
RainerConst == E*BenoitConst
Out[5]=
True  (* what else ? *)

Greetings,
Peter

Rainer Rosenthal schrieb:
> Thanks to Neil's latest update I am proud to
> announce A136616 and A136617, dealing with
> harmonic numbers.
> 
> ...  
> 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);
the error is here:
> 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...
...
> 
> Best regards,
> Rainer Rosenthal
> r.rosenthal at web.de
> 
> 
>