A222068 Decimal expansion of (1/16)*Pi^2.

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Feb 10 2013

Keywords

• nonn
• cons

Comments

Conjectured to be density of densest packing of equal spheres in four dimensions (achieved for example by the D_4 lattice).

From Hugo Pfoertner, Aug 29 2018: (Start)

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).

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 ...".

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...

(End)

From Robert Israel, Aug 31 2018: (Start)

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.

Sum_{k>=0} (-1)^k d(2k+1)/(2k+1)

= Sum_{k>=0} Sum_{2i+1 | 2k+1} (-1)^k/(2k+1)

(letting 2k+1=(2i+1)(2j+1): note that k == i+j (mod 2))

= Sum_{i>=0} Sum_{j>=0} (-1)^(i+j)/((2i+1)(2j+1))

= (Sum_{i>=0} (-1)^i/(2i+1))^2 = (Pi/4)^2. (End)

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

Examples

			0.6168502750680849136771556874922594459571...
		

References

• S. D. Chowla, Solution and Remarks on Question 770, J. Indian Math. Soc. 17 (1927-28), 166-171.
• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 507.
• S. Ramanujan, Coll. Papers, Chelsea, 1962, Question 770, page 333.
• G. N. Watson, Solution to Question 770, J. Indian Math. Soc. 18 (1929-30), 294-298.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• B. C. Berndt, Y. S. Choi and S. Y. Kang, The problems submitted by Ramanujan to the Journal of Indian Math. Soc., in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q770, JIMS VIII).
• J. H. Conway and N. J. A. Sloane, What are all the best sphere packings in low dimensions?, Discr. Comp. Geom., 13 (1995), 383-403.
• Mathematics StackExchange, Sum_k (-1)^k tau(2k+1)/(2k+1).
• G. Nebe and N. J. A. Sloane, Home page for D_4 lattice.
• N. J. A. Sloane and Andrey Zabolotskiy, Table of maximal density of a packing of equal spheres in n-dimensional Euclidean space (some values are only conjectural).
• N. J. Wildberger, Divine Proportions: Rational Trigonometry to Universal Geometry, Wild Egg Books, Sydney 2005.
• Index entries for transcendental numbers.

Crossrefs

Cf. A000005.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222069, A222070, A222071, A222072, A260646.

Programs

Magma

pi:=Pi(RealField(110)); Reverse(Intseq(Floor((1/16)*10^100*pi^2))); // Vincenzo Librandi, Feb 20 2017

Mathematica

RealDigits[N[Gamma[3/2]^4, 104]] (* Fred Daniel Kline, Feb 19 2017 *)
RealDigits[N[Pi^2/16, 100]][[1]] (* Vincenzo Librandi, Feb 20 2017 *)
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 *)

PARI

(Pi/4)^2 \\ Charles R Greathouse IV, Oct 31 2014

Formula

Equals A003881^2. - Bruno Berselli, Feb 11 2013

Equals A123092+1/2. - R. J. Mathar, Feb 15 2013

Equals Integral_{x>0} x^2*log(x)/((1+x)^2*(1+x^2)) dx. - Jean-François Alcover, Apr 29 2013

Equals the Bessel moment integral_{x>0} x*I_0(x)*K_0(x)^3. - Jean-François Alcover, Jun 05 2016

Equals Sum_{k>=1} zeta(2*k)*k/4^k. - Amiram Eldar, May 29 2021