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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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8
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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
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OFFSET
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1,1
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COMMENTS
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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
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 9.
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LINKS
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Eric Weisstein's World of Mathematics, Ball.
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FORMULA
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d = root of Psi((1/2)*d + 1) = log(Pi).
d is 2 less than the number with decimal digits A074457 (the hypersphere dimension that maximizes hypersurface area). - Eric W. Weisstein, Dec 02 2014
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EXAMPLE
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5.256946404860576780132838388690769236619017237183214857509879678777109...
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MATHEMATICA
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x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105] // RealDigits // First (* Jean-François Alcover, Mar 28 2013 *)
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PROG
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(PARI)
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)
/* David W. Cantrell */
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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