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A074455
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Consider volume of unit sphere as a function of the dimension d; maximize this as a function of d (considered as a continuous variable); sequence gives decimal expansion of the best d.
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5
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5, 2, 5, 6, 9, 4, 6, 4, 0, 4, 8, 6, 0, 5, 7, 6, 7, 8, 0, 1, 3, 2, 8, 3, 8, 3, 8, 8, 6, 9, 0, 7, 6, 9, 2, 3, 6, 6, 1, 9, 0, 1, 7, 2, 3, 7, 1, 8, 3, 2, 1, 4, 8, 5, 7, 5, 0, 9, 8, 7, 9, 6, 7, 8, 7, 7, 7, 1, 0, 9, 3, 4, 6, 7, 3, 6, 8, 2, 0, 2, 7, 2, 8, 1, 7, 7, 2, 0, 2, 3, 8, 4, 8, 9, 7, 9, 2, 4, 6, 9, 2, 6
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Comments from David W. Wilson (davidwwilson(AT)comcast.net), Jul 12 2007: (Start) For an integer d, the volume of a d-dimensional unit ball is v(d) = pi^(d/2)/(d/2)! and its surface area is area(d) = d pi^(d/2)/(d/2)! = d v(d). If we interpolate n! = gamma(n+1) we can define v(d) and area(d) as continuous functions for (at least) d >= 0.
A074457 purports to minimize area(d). Since area(d+2) = 2 pi v(d), area() is minimized at y = x+2, therefore A074457 coincides with the current sequence except at the first term. (End)
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REFERENCES
| J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.
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LINKS
| Eric Weisstein's World of Mathematics, Ball
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FORMULA
| d = root of Psi(1/2 d + 1) = log(Pi).
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EXAMPLE
| 5.2569464048605767801328383886907692366190172371832148575098796787771093\
4673682027281772023848979246926957...
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PROG
| /* 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
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CROSSREFS
| Cf. A074457.
The volume is given by A074454. Cf. A072345 & A072346.
Sequence in context: A196626 A082571 A087300 * A142702 A201530 A085997
Adjacent sequences: A074452 A074453 A074454 * A074456 A074457 A074458
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KEYWORD
| cons,nonn
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AUTHOR
| Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 22 2002
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EXTENSIONS
| Corrected by Eric Weisstein (eric(AT)weisstein.com), Aug 31, 2003 and by Martin Fuller (martin_n_fuller(AT)btinternet.com), Jul 12 2007
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