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%I #37 Feb 21 2023 10:34:37
%S 1,9,7,3,9,2,0,8,8,0,2,1,7,8,7,1,7,2,3,7,6,6,8,9,8,1,9,9,9,7,5,2,3,0,
%T 2,2,7,0,6,2,7,3,9,8,8,1,4,4,8,1,5,8,1,2,5,2,8,2,6,6,9,8,7,5,2,4,4,0,
%U 0,8,9,6,4,4,8,3,8,4,1,0,4,8,6,0,0,3,5,4,6,8,0,7,4,3,7,1,0,4,4,6,3,6,4,8,0
%N Decimal expansion of 2*Pi^2.
%C Surface area of the 4-dimensional unit sphere. The volume of the 4-dimensional unit sphere is a fourth of this, A102753.
%C Also decimal expansion of Pi^2/5 = 1.973920..., with offset 1. - _Omar E. Pol_, Oct 04 2011
%D L. A. Santalo, Integral Geometry and Geometric Probability, Addison-Wesley, 1976, see p. 15.
%H G. C. Greubel, <a href="/A164102/b164102.txt">Table of n, a(n) for n = 2..5000</a>
%H Yann Bernard, <a href="http://images.math.cnrs.fr/Autour-des-surfaces-de-Willmore.html">Autour des surfaces de Willmore</a>, Images des Mathématiques, CNRS, 2014 (in French).
%H Fernando C. Marques and André Neves, <a href="https://arxiv.org/abs/1202.6036">Min-Max theory and the Willmore conjecture</a>, arXiv:1202.6036 [math.DG], 2012-2013.
%H H.-J. Seiffert, <a href="https://fq.math.ca/Scanned/29-4/elementary29-4.pdf">Problem B-705</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 29, No. 4 (1991), p. 372; <a href="https://www.fq.math.ca/Scanned/31-1/elementary31-1.pdf">An Application of a Series Expansion for (arcsinx)^2</a>, Solution to Problem B-705, ibid., Vol. 31, No. 1 (1993), pp. 85-86.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hypersphere.html">Hypersphere</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypersphere">Hypersphere</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Willmore_conjecture">Willmore conjecture</a>.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals 2*A002388 = 4*A102753.
%F Pi^2/5 = Sum_{k>=1} Lucas(2*k)/(k^2*binomial(2*k,k)) = Sum_{k>=1} A005248(k)/A002736(k) (Seiffert, 1991). - _Amiram Eldar_, Jan 17 2022
%e 19.739208802178717237668981...
%t RealDigits[2*Pi^2,10,120][[1]] (* _Harvey P. Dale_, Apr 19 2012 *)
%o (PARI) 2*Pi^2 \\ _Charles R Greathouse IV_, Jan 24 2014
%Y Cf. A000032, A002388, A002736, A005248, A091476, A013661.
%K cons,nonn,easy
%O 2,2
%A _R. J. Mathar_, Aug 10 2009
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