3 seqs that should be the same but are not

[David W. Cantrell]

Martin,

Concerning your question "Can someone explain why the dimension with 
maximum surface area is exactly 2 more than the dimension with maximum 
volume?":

On a "mechanical" level, explaining why is trivial. If we let invdig 
denote the appropriate inverse of the digamma function, then we can 
solve the required transcendental equations to obtain

2 invdig(log(pi))

and

2 invdig(log(pi)) + 2.

But you were probably wanting something deeper, and I'm not sure what 
that is.

David

----- Original Message ----- 
From: "Martin Fuller" <martin_n_fuller at btinternet.com>
To: "David W. Cantrell" <DWCantrell at sigmaxi.net>; "Jonathan Post" 
<jvospost3 at gmail.com>; <njas at research.att.com>
Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>; 
"jonathan post" <jvospost2 at yahoo.com>
Sent: Thursday, July 12, 2007 14:53
Subject: Re: 3 seqs that should be the same but are not


> A074455 and A087300 should be the same: A074455 has the right 
> description and
> came first, A087300 has the right values.
>
> I get a different value to OEIS for A074457 as well as for A074455.
>
> See below for the current values and my calculations in PARI.  The 
> PARI code
> gives the same value as A087300 and is consistent when increasing 
> the
> precision, so I think it's right.
>
> Martin Fuller
>
> PS Can someone explain why the dimension with maximum surface area 
> is exactly 2
> more than the dimension with maximum volume?
>
> PPS All these constants are based on geometers dimension which is 
> one more than
> topologists dimension (see the comments at MathWorld and compare 
> with the
> formulae at PlanetMath
> http://planetmath.org/encyclopedia/VolumeOfTheNSphere.html)
>
> OEIS
> A087300
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987966544135040807732427416016036408330066793834
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987966581295771725141939748631782716397233020121
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> PARI/GP
> A074455
> 5.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074457
> 7.25694640486057678013283838869076923661901723718321485750987967877710934673682027281772023848979246926957
> A074454
> 5.27776802111340099728214586417284638752999928451017356776163734021486412730547017110062048407258401284645
> A074456
> 33.1611944849620026918630240155829735800472328410872585131001181554037565464718434466607460949351387694776
>
> /* PARI/GP code */
> hyperspheresurface(d)=2*Pi^(d/2)/gamma(d/2)
> hyperspherevolume(d)=hyperspheresurface(d)/d
>
> FindMax(fn_x,lo,hi)=
> {
>  local(oldprecision, x, y, z);
>  oldprecision = default(realprecision);
>  default(realprecision, oldprecision+10);
>
>  while (hi-lo > 10^-oldprecision,
>    while (1,
>      z = vector(2, i, lo*(3-i)/3 + hi*i/3);
>      y = vector(2, i, eval(Str("x = z[" i "]; " fn_x)));
>      if (abs(y[1]-y[2]) > 10^(5-default(realprecision)), break);
>      default(realprecision, default(realprecision)+10);
>    );
>    if (y[1] < y[2], lo = z[1], hi = z[2]);
>  );
>
>  default(realprecision, oldprecision);
>  (lo + hi) / 2.
> }
>
> default(realprecision, 105);
> A074455=FindMax("hyperspherevolume(x)", 1, 9)
> A074457=FindMax("hyperspheresurface(x)", 1, 9)
> A074454=hyperspherevolume(A074455)
> A074456=hyperspheresurface(A074457)
> /* PARI/GP code ends */
>
> --- "David W. Cantrell" <DWCantrell at sigmaxi.net> wrote:
>
>> ----- Original Message ----- 
>> From: "Jonathan Post" <jvospost3 at gmail.com>
>> To: <njas at research.att.com>
>> Cc: <seqfan at ext.jussieu.fr>; <eric at weisstein.com>; <rgwv at rgwv.com>;
>> "jonathan post" <jvospost2 at yahoo.com>
>> Sent: Thursday, July 12, 2007 13:12
>> Subject: Re: 3 seqs that should be the same but are not
>>
>>
>> > Why should "surface area" (A074457) equal "volume" (A087300)?
>>
>> They're not the same constant, but the constant given in A074457 is
>> simply 2 more than the constant given in A087300. Both entries 
>> should
>> mention that, I think.
>>
>> There are also two errors in the title of A087300. It should read:
>> Decimal expansion of value of d for which volume of d-dimensional 
>> unit
>> ball is maximized.
>> [Note that it is necessary to specify that it's a _unit_ ball.]
>>
>> David
>>
>>