3 seqs that should be the same but are not

[Martin Fuller]

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 
> 
>