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

0,1

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

References

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. xix.

S. D. Chowla, Solution and Remarks on Question 770, J. Indian Math. Soc. 17 (1927-28), 166-171.

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.

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

Example

0.6168502750680849136771556874922594459571...

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 *)

Prog

(PARI) (Pi/4)^2 \\ Charles R Greathouse IV, Oct 31 2014
(Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor((1/16)*10^100*pi^2))); // Vincenzo Librandi, Feb 20 2017

Crossrefs

Cf. A000005.

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

Sequence in context: A156163 A301817 A011300 * A272055 A157292 A159828

Adjacent sequences: A222065 A222066 A222067 * A222069 A222070 A222071

Keyword

nonn,cons

Author

N. J. A. Sloane, Feb 10 2013

Status

approved