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%I #78 Dec 30 2023 10:05:01
%S 6,1,6,8,5,0,2,7,5,0,6,8,0,8,4,9,1,3,6,7,7,1,5,5,6,8,7,4,9,2,2,5,9,4,
%T 4,5,9,5,7,1,0,6,2,1,2,9,5,2,5,4,9,4,1,4,1,5,0,8,3,4,3,3,6,0,1,3,7,5,
%U 2,8,0,1,4,0,1,2,0,0,3,2,7,6,8,7,6,1,0,8,3,7,7,3,2,4,0,9,5,1,4,4,8,9,0,0
%N Decimal expansion of (1/16)*Pi^2.
%C Conjectured to be density of densest packing of equal spheres in four dimensions (achieved for example by the D_4 lattice).
%C From _Hugo Pfoertner_, Aug 29 2018: (Start)
%C Also decimal expansion of Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1), where d(n) is the number of divisors of n A000005(n).
%C Ramanujan's question 770 in the Journal of the Indian Mathematical Society (VIII, 120) asked "If d(n) denotes the number of divisors of n, show that d(1) - d(3)/3 + d(5)/5 - d(7)/7 + d(9)/9 - ... is a convergent series ...".
%C A summation of the first 2*10^9 terms performed by _Hans Havermann_ yields 0.6168503077..., which is close to (Pi/4)^2=0.616850275...
%C (End)
%C From _Robert Israel_, Aug 31 2018: (Start)
%C Modulo questions about rearrangement of conditionally convergent series, which I expect a more careful treatment would handle, Sum_{k>=0} (-1)^k*d(2*k+1)/(2*k+1) should indeed be Pi^2/16.
%C Sum_{k>=0} (-1)^k d(2k+1)/(2k+1)
%C = Sum_{k>=0} Sum_{2i+1 | 2k+1} (-1)^k/(2k+1)
%C (letting 2k+1=(2i+1)(2j+1): note that k == i+j (mod 2))
%C = Sum_{i>=0} Sum_{j>=0} (-1)^(i+j)/((2i+1)(2j+1))
%C = (Sum_{i>=0} (-1)^i/(2i+1))^2 = (Pi/4)^2. (End)
%C Volume bounded by the surface (x+y+z)^2-2(x^2+y^2+z^2)=4xyz, the ellipson (see Wildberger, p. 287). - _Patrick D McLean_, Dec 03 2020
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
%D S. D. Chowla, Solution and Remarks on Question 770, J. Indian Math. Soc. 17 (1927-28), 166-171.
%D S. Ramanujan, Coll. Papers, Chelsea, 1962, Question 770, page 333.
%D G. N. Watson, Solution to Question 770, J. Indian Math. Soc. 18 (1929-30), 294-298.
%H Vincenzo Librandi, <a href="/A222068/b222068.txt">Table of n, a(n) for n = 0..10000</a>
%H B. C. Berndt, Y. S. Choi and S. Y. Kang, <a href="https://www.researchgate.net/publication/2575787">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q770, JIMS VIII).
%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/BF02574051">What are all the best sphere packings in low dimensions?</a>, Discr. Comp. Geom., 13 (1995), 383-403.
%H Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/2903015/sum-k-1k-frac-tau2k12k1">Sum_k (-1)^k tau(2k+1)/(2k+1)</a>.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D4.html">Home page for D_4 lattice</a>.
%H N. J. A. Sloane and Andrey Zabolotskiy, <a href="/A093825/a093825_1.txt">Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural)</a>.
%H N. J. Wildberger, <a href="https://www.researchgate.net/publication/266738365">Divine Proportions: Rational Trigonometry to Universal Geometry</a>, Wild Egg Books, Sydney 2005.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F Equals A003881^2. - _Bruno Berselli_, Feb 11 2013
%F Equals A123092+1/2. - _R. J. Mathar_, Feb 15 2013
%F Equals Integral_{x>0} x^2*log(x)/((1+x)^2*(1+x^2)) dx. - _Jean-François Alcover_, Apr 29 2013
%F Equals the Bessel moment integral_{x>0} x*I_0(x)*K_0(x)^3. - _Jean-François Alcover_, Jun 05 2016
%F Equals Sum_{k>=1} zeta(2*k)*k/4^k. - _Amiram Eldar_, May 29 2021
%e 0.6168502750680849136771556874922594459571...
%t RealDigits[N[Gamma[3/2]^4, 104]] (* _Fred Daniel Kline_, Feb 19 2017 *)
%t RealDigits[N[Pi^2/16, 100]][[1]] (* _Vincenzo Librandi_, Feb 20 2017 *)
%t Integrate[Boole[(x+y+z)^2-2(x^2+y^2+z^2)>4x y z],{x,0,1},{y,0,1},{z,0,1}] (* _Patrick D McLean_, Dec 03 2020 *)
%o (PARI) (Pi/4)^2 \\ _Charles R Greathouse IV_, Oct 31 2014
%o (Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor((1/16)*10^100*pi^2))); // _Vincenzo Librandi_, Feb 20 2017
%Y Cf. A000005.
%Y Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222069, A222070, A222071, A222072, A260646.
%K nonn,cons
%O 0,1
%A _N. J. A. Sloane_, Feb 10 2013
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