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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 resulting volume.
3

%I #14 Jan 21 2025 15:26:41

%S 5,2,7,7,7,6,8,0,2,1,1,1,3,4,0,0,9,9,7,2,8,2,1,4,5,8,6,4,1,7,2,8,4,6,

%T 3,8,7,5,2,9,9,9,9,2,8,4,5,1,0,1,7,3,5,6,7,7,6,1,6,3,7,3,4,0,2,1,4,8,

%U 6,4,1,2,7,3,0,5,4,7,0,1,7,1,1,0,0,6,2,0,4,8,4,0,7,2,5,8,4,0,1

%N 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 resulting volume.

%C The dimension is given in A074455.

%C If you set v(n) = Pi^(n/2)/(n/2)! and s(n) = n*Pi^(n/2)/(n/2)! and then plot {6.283*v(n-2), s(n)} for 0<=n<=20, the two curves are almost identical.

%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 67.

%e 5.277768021113400997282145864172846387529999284510173567761637340214864\

%e 12730547017110062048407258401284645...

%t d = x /. FindRoot[ PolyGamma[1 + x/2] == Log[Pi], {x, 5}, WorkingPrecision -> 105]; First[ RealDigits[ Pi^(d/2)/(d/2)!]][[1 ;; 99]] (* _Jean-François Alcover_, Apr 12 2013 *)

%Y Cf. A072345, A072346.

%K cons,nonn

%O 1,1

%A _Robert G. Wilson v_, Aug 22 2002

%E Checked by _Martin Fuller_, Jul 12 2007