A222072 -id:A222072 - cp's OEIS frontend

A007331 Fourier coefficients of E_{infinity,4}.

0, 1, 8, 28, 64, 126, 224, 344, 512, 757, 1008, 1332, 1792, 2198, 2752, 3528, 4096, 4914, 6056, 6860, 8064, 9632, 10656, 12168, 14336, 15751, 17584, 20440, 22016, 24390, 28224, 29792, 32768, 37296, 39312, 43344, 48448, 50654, 54880, 61544, 64512

Offset: 0

N. J. A. Sloane, Mira Bernstein

E_{infinity,4} is the unique normalized weight-4 modular form for Gamma_0(2) with simple zeros at i*infinity. Since this has level 2, it is not a cusp form, in contrast to A002408.
a(n+1) is the number of representations of n as a sum of 8 triangular numbers (from A000217). See the Ono et al. link, Theorem 5.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) gives the sum of cubes of divisors d of n such that n/d is odd. This is called sigma^#3(n) in the Ono et al. link. See a formula below. - _Wolfdieter Lang, Jan 12 2017

			G.f. = q + 8*q^2 + 28*q^3 + 64*q^4 + 126*q^5 + 224*q^6 + 344*q^7 + 512*q^8 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 139, Ex (ii).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1001 from T. D. Noe)
B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3, 257-274.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 187.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94. Theorem 5.
H. Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
Index entries for sequences mentioned by Glaisher.

Cf. A002408, A004017, A035016, A045825, A076577, A096960, A222072.

Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809, A076577.

Basis( ModularForms( Gamma0(2), 4), 10) [2]; /* Michael Somos, May 27 2014 */

nmax:=40: seq(coeff(series(x*(product((1-x^k)^8*(1+x^k)^16, k=1..nmax)), x, n+1), x, n), n=0..nmax); # Vaclav Kotesovec, Oct 14 2015

Prepend[Table[Plus @@ (Select[Divisors[k + 1], OddQ[(k + 1)/#] &]^3), {k, 0, 39}], 0] (* Ant King, Dec 04 2010 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q^(1/2)]^8 / 256, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := If[ n < 1, 0, Sum[ d^3 Boole[ OddQ[ n/d]], {d, Divisors[ n]}]]; (* Michael Somos, Jun 04 2013 *)
f[n_] := Total[(2n/Select[ Divisors[ 2n], Mod[#, 4] == 2 &])^3]; Flatten[{0, Array[f, 40] }] (* Robert G. Wilson v, Mar 26 2015 *)
nmax=60; CoefficientList[Series[x*Product[(1-x^k)^8 * (1+x^k)^16, {k,1,nmax}],{x,0,nmax}], x] (* Vaclav Kotesovec, Oct 14 2015 *)
QP = QPochhammer; s = q * (QP[-1, q]/2)^16 * QP[q]^8 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8, n))}; /* Michael Somos, May 31 2005 */

a(n)=my(e=valuation(n,2)); 8^e * sigma(n/2^e, 3) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors
def a(n):
    return 0 if n == 0 else sum(((n//d)%2)*d**3 for d in divisors(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

ModularForms( Gamma0(2), 4, prec=33).1; # Michael Somos, Jun 04 2013

G.f.: q * Product_{k>=1} (1-q^k)^8 * (1+q^k)^16. - corrected by Vaclav Kotesovec, Oct 14 2015

a(n) = Sum_{0

G.f.: Sum_{n>0} n^3*x^n/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002

Expansion of Jacobi theta constant theta_2(q)^8 / 256 in powers of q.

Expansion of eta(q^2)^16 / eta(q)^8 in powers of q. - Michael Somos, May 31 2005

Expansion of x * psi(x)^8 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jan 15 2012

Expansion of (Q(x) - Q(x^2)) / 240 in powers of x where Q() is a Ramanujan Lambert series. - Michael Somos, Jan 15 2012

Expansion of E_{gamma,2}^2 * E_{0,4} in powers of q.

Euler transform of period 2 sequence [8, -8, ...]. - Michael Somos, May 31 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - u^2*w + 16*u*v*w - 32*v^2*w + 256*v*w^2. - Michael Somos, May 31 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 16^(-1) (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035016. - Michael Somos, Jan 11 2009

Multiplicative with a(2^e) = 2^(3e), a(p^e) = (p^(3(e+1))-1)/(p^3-1). - Mitch Harris, Jun 13 2005

Dirichlet convolution of A154955 by A001158. Dirichlet g.f. zeta(s)*zeta(s-3)*(1-1/2^s). - R. J. Mathar, Mar 31 2011

A002408(n) = -(-1)^n * a(n).

Convolution square of A008438. - Michael Somos, Jun 15 2014

a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017

Sum_{k=1..n} a(k) ~ c * n^4, where c = Pi^4/384 = 0.253669... (A222072). - Amiram Eldar, Oct 19 2022

Additional comments from Barry Brent (barryb(AT)primenet.com)
Wrong Maple program replaced by Vaclav Kotesovec, Oct 14 2015
a(0)=0 prepended by Vaclav Kotesovec, Oct 14 2015

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

Original entry on oeis.org

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

N. J. A. Sloane, Feb 10 2013

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

			0.6168502750680849136771556874922594459571...

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.

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.

Cf. A000005.

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

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

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 *)

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

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

A020769 Decimal expansion of 1/sqrt(12) = 1/(2*sqrt(3)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Center density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
Let a equal the length of one side of an equilateral triangle and let b equal the radius of the circle inscribed in that triangle. This sequence gives the decimal expansion of b/a. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Feb 20 2004
The constant (3+sqrt 3)/6, which is 0.5 larger than this, plays a role in Borsuk's conjecture. - Arkadiusz Wesolowski, Mar 17 2014

			0.28867513459481288225457439025097872782380087563506343800930116324198883615...

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
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.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
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).
Wikipedia, Borsuk's conjecture
Index entries for algebraic numbers, degree 2

Related constants: A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

RealDigits[N[1/Sqrt[12],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

1/sqrt(12) \\ Charles R Greathouse IV, Oct 31 2014

A093825 Decimal expansion of Pi/(3*sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 16 2004

Density of densest packing of equal spheres in three dimensions (achieved for example by the fcc lattice).

Atomic packing factor (APF) of the face-centered-cubic (fcc) and the hexagonal-close-packed (hcp) crystal lattices filled with spheres of the same diameter. - Stanislav Sykora, Sep 29 2014

			0.74048048969306104116931349834344894976910361489594837...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer, 3rd. ed., 1998. See p. 15, line n = 3.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.7, p. 506.
Clifford A. Pickover, The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics (2009), at p. 126.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Harry J. Smith, Table of n, a(n) for n = 0..20000
James Grime and Brady Haran, The Best Way to Pack Spheres, Numberphile video (2018).
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.
Thomas C. Hales, Dense Sphere Packings, Cambridge University Press, 2012.
G. Nebe and N. J. A. Sloane, Home page for fcc 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).
Eric Weisstein's World of Mathematics, Cubic Close Packing.
Eric Weisstein's World of Mathematics, Ellipsoid Packing.
Eric Weisstein's World of Mathematics, Sphere Packing.
Wikipedia, Atomic packing factor.
Index entries for sequences related to f.c.c. lattice.
Index entries for transcendental numbers.

Cf. A093824.
Cf. APF's of other crystal lattices: A019673 (simple cubic), A247446 (diamond cubic).
Cf. A161686 (continued fraction).
Related constants: A020769, A020789, A093766, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

RealDigits[Pi/(3 Sqrt[2]), 10, 120][[1]] (* Harvey P. Dale, Feb 03 2012 *)

default(realprecision, 20080); x=10*Pi*sqrt(2)/6; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b093825.txt", n, " ", d)); \\ Harry J. Smith, Jun 18 2009

Pi/sqrt(18) \\ Charles R Greathouse IV, May 11 2017

Equals A019670*A010503. - R. J. Mathar, Feb 05 2009

Equals Integral_{x >= 0} (4*x^2 + 1)/((2*x^2 + 1)*(8*x^2 + 1)) dx. - Peter Bala, Feb 12 2025

Entry revised by N. J. A. Sloane, Feb 10 2013

A260646 Decimal expansion of Pi^12/12!, the absolute density of the Leech lattice.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, Nov 12 2015

			0.001929574309403923047903345563685957640168471815...

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Maryna Viazovska, The sphere packing problem in dimension 24, Annals of Mathematics, 185 (2017), 1017-1033; arXiv:1603.06518 [math.NT], 2016-2017.
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1998 (see also Corrections and Updates), Chapter 4, Section 11.
G. Nebe and N. J. A. Sloane, Home page for Leech lattice.
A. Roberts, Properties of the Leech Lattice, 2006.
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).
Wikipedia, Leech lattice
Index entries for transcendental numbers

Densities of other lattices: A093766, A093825, A222068, A222069, A222070, A222071, A222072.

Related to Leech lattice: A008408, A323282.

RealDigits[N[Pi^12/12!, 120]]//First (* Michael De Vlieger, Nov 12 2015 *)

{ default(realprecision, 50080); x=Pi^12/12!; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) }

A222066 Decimal expansion of 1/sqrt(128).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be center density of densest packing of equal spheres in five dimensions (achieved for example by the D_5 lattice).

			.088388347648318440550105545263106129910604492211...

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

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.
G. Nebe and N. J. A. Sloane, Home page for D_5 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).
Index entries for algebraic numbers, degree 2

Related constants: A020769, A020789, A093766, A093825, A222067, A222068, A222069, A222070, A222071, A222072, A260646.

Join[{0},RealDigits[1/Sqrt[128],10,120][[1]]] (* Harvey P. Dale, Sep 20 2023 *)

1/sqrt(128) \\ Charles R Greathouse IV, Oct 31 2014

Equals A020789/2. - R. J. Mathar, Jan 27 2021

A222070 Decimal expansion of (1/144)3^(1/2)Pi^3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in six dimensions (achieved for example by the E_6 lattice).

			0.3729475455820649395634775586799581063936647972683873631...

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.

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.
G. Nebe and N. J. A. Sloane, Home page for E_6 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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222071, A222072, A260646.

RealDigits[Sqrt[3]*Pi^3/144, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Pi^3*sqrt(3)/144 \\ Charles R Greathouse IV, Oct 31 2014

A222071 Decimal expansion of (1/105)*Pi^3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in 7 dimensions (achieved for example by the E_7 lattice).

			0.295297873145712573099774429210489478116431319675...

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.

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.
G. Nebe and N. J. A. Sloane, Home page for E_7 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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222072, A260646.

RealDigits[Pi^3/105, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Pi^3/105 \\ Charles R Greathouse IV, Oct 31 2014

A222069 Decimal expansion of (1/30)2^(1/2)Pi^2.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in five dimensions (achieved for example by the D_5 lattice).

			.46525761330925863561050406241129368599465775139653615774...

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.

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.
G. Nebe and N. J. A. Sloane, Home page for D_5 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).
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222070, A222071, A222072, A260646.

RealDigits[(Sqrt[2] Pi^2)/30,10,120][[1]] (* Harvey P. Dale, Nov 07 2021 *)

Pi^2/sqrt(450) \\ Charles R Greathouse IV, Oct 31 2014

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cp's OEIS Frontend

A093766 Decimal expansion of Pi/(2*sqrt(3)).

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A007331 Fourier coefficients of E_{infinity,4}.

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

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A020769 Decimal expansion of 1/sqrt(12) = 1/(2*sqrt(3)).

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A093825 Decimal expansion of Pi/(3*sqrt(2)).

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A260646 Decimal expansion of Pi^12/12!, the absolute density of the Leech lattice.

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A222070 Decimal expansion of (1/144)3^(1/2)Pi^3.

A222069 Decimal expansion of (1/30)2^(1/2)Pi^2.