A093766 -id:A093766 - cp's OEIS frontend

A121361 Expansion of f(x^1, x^5) * psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.

1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 2, 1, 0, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 2, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 0, 0, 1, 0, 1, 0, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 0, 2, 2, 1, 3, 0, 0, 0, 1, 0, 0

Offset: 0

Michael Somos, Jul 16 2006

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + 2*x^7 + x^8 + x^10 + x^11 + ...
G.f. = q^7 + q^19 + q^31 + q^43 + q^67 + q^79 + 2*q^91 + q^103 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A093766, A093829.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^6] QPochhammer[ -x^5, x^6] QPochhammer[ x^6] EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}]; (* Michael Somos, Sep 02 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)), n))};

Expansion of q^(-7/12) * eta(q^2) * eta(q^3) * eta(q^4) * eta(q^12) /
(eta(q) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 1, 0, 0, -1, 1, 0, 1, -1, 0, 0, 1, -2, ...].
2*a(n) = A093829(12*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 20 2025

A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0..oo.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 25 2011

Reduction of the integral by Robert Israel, Jul 25 2012: (Start)
Use the definition of DedekindEta as a sum:
Eta(i*x) = Sum_{n=-oo..oo} (-1)^n*exp(-Pi*x*(6n-1)^2/12).
Now Integral_{x=0..oo} exp(-Pi*x*(6n-1)^2/12) dx = 12/(Pi*(6n-1)^2).
According to Maple, Sum_{n=-oo..oo} (-1)^n*12/(Pi*(6n-1)^2) is
2*3^(1/2)*(dilog(1-(1/2)*i-(1/2)*3^(1/2)) - dilog(1-(1/2)*i+(1/2)*3^(1/2)) - dilog(1+(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)))/Pi
(Jonquiere's inversion formula -- see http://en.wikipedia.org/wiki/Polylogarithm)
(but note that Maple's dilog(z) is L_2(1-z) in the notation there) gives
dilog(1-(1/2)*i-(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)) = (13/72)*Pi^2
and
dilog(1-(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i+(1/2)*3^(1/2)) = -11*Pi^2/72
which give the desired multiple of Pi. (End)
Ratio of surface area of a sphere to the regular octahedron whose edge equals the radius of the sphere. - Omar E. Pol, Dec 30 2023

			3.627598728468435701188156515284311464568132496185481151139769870776...

Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.

D. H. Lehmer, Interesting series involving the central binomial coefficient, Am. Math. Monthly 92 (7) (1985) 449.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 511.
Eric W. Weisstein's World of Mathematics, Dedekind Eta Function.

Cf. A073005, A073006, A093602, A093766,

RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* Robert G. Wilson v, Nov 18 2012 *)

intnum(x=1e-7,1e6,eta(x*I,1)) \\ Charles R Greathouse IV, Feb 26 2011

Equals 2*Pi/sqrt(3), 2 times A093602, and in consequence equal to Sum_{m>=1} 3^m/(m*binomial(2m,m)) according to Lehmer. - R. J. Mathar, Jul 24 2012
Also equals Gamma(1/3)*Gamma(2/3) = A073005 * A073006. - Jean-François Alcover, Nov 24 2014
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.
Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)
Equals Integral_{x=0..2*Pi} 1/(2 + sin(x)) dx; since for a>1: Integral_{x=0..2*Pi} 1/(a + sin(x)) dx = 2*Pi/sqrt(a^2-1). - Bernard Schott, Feb 18 2023
Equals 4*A093766. - Omar E. Pol, Dec 30 2023
From Stefano Spezia, Jun 05 2025: (Start)
Equals Beta(1/3,2/3).
Equals Integral_{x=-oo..oo} 1/(x^2 + x + 1) dx.
Equals 2*Integral_{x=0..oo} log(1 + x^3)/x^3 dx.
Equals Integral_{x=0..oo} log(1 + 4/(x*(x + 2))) dx. (End)

A134667 Period 6: repeat [0, 1, 0, 0, 0, -1].

Original entry on oeis.org

0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0

Offset: 0

Paul Curtz, Jan 26 2008

Dirichlet series for the non-principal character modulo 6: L(s,chi) = Sum_{n>=1} a(n)/n^s. For example L(1,chi) = A093766, L(2,chi) = A214552, and L(3,chi) = Pi^3/(18*sqrt(3)). See Jolley eq. (314) and arXiv:1008.2547 L(m=6,r=2,s). - R. J. Mathar, Jul 31 2010

			G.f. = x - x^5 + x^7 - x^11 + x^13 - x^17 + x^19 - x^23 + x^25 - x^29 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=6, Chi_2(n).
L. B. W. Jolley, Summation of Series, Dover (1961).

R. J. Mathar, Table of Dirichlet L-series.., arXiv:1008.2547 [math.NT], 2010-2015.
Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1).

Cf. A093766, A120325, A131719, A131720, A131735, A131736, A214552.

&cat[[0, 1, 0, 0, 0, -1]^^20]; // Wesley Ivan Hurt, Jun 20 2016

A134667:=n->[0, 1, 0, 0, 0, -1][(n mod 6)+1]: seq(A134667(n), n=0..100);
# Wesley Ivan Hurt, Jun 20 2016

a[ n_] := JacobiSymbol[-12, n]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := {1, 0, 0, 0, -1, 0}[[Mod[n, 6, 1]]]; (* Michael Somos, Apr 24 2014 *)
PadRight[{},120,{0,1,0,0,0,-1}] (* Harvey P. Dale, Aug 01 2021 *)

{a(n) = [0, 1, 0, 0, 0, -1][n%6+1]}; /* Michael Somos, Feb 10 2008 */

{a(n) = kronecker(-12, n)}; /* Michael Somos, Feb 10 2008 */

{a(n) = if( n < 0, -a(-n), if( n<1, 0, direuler(p=2, n, 1 / (1 - kronecker(-12, p) * X))[n]))}; /* Michael Somos, Aug 11 2009 */

Euler transform of length 6 sequence [0, 0, 0, -1, 0, 1]. - Michael Somos, Feb 10 2008
G.f.: x * (1 - x^4) / (1 - x^6) = x*(1+x^2) / (1 + x^2 + x^4) = x*(1+x^2) / ( (1+x+x^2)*(x^2-x+1) ).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3)) where f(u, v, w) = w * (2 + v - u^2 - 2*v^2) - 2 * u * v. - Michael Somos, Aug 11 2009
a(n) is multiplicative with a(p^e) = 0^e if p = 2 or p = 3, a(p^e) = 1 if p == 1 (mod 6), a(p^e) = (-1)^e if p == 5 (mod 6). - Michael Somos, Aug 11 2009
a(-n) = -a(n). a(n+6) = a(n). a(2*n) = a(3*n) = 0.
sqrt(3)*a(n) = sin(Pi*n/3) + sin(2*Pi*n/3). - R. J. Mathar, Oct 08 2011
a(n) + a(n-2) + a(n-4) = 0 for n>3. - Wesley Ivan Hurt, Jun 20 2016
E.g.f.: 2*sin(sqrt(3)*x/2)*cosh(x/2)/sqrt(3). - Ilya Gutkovskiy, Jun 21 2016

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, ", ")) }

A035178 a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 0, 1, 0, 1, 2, 0, 2, 0, 0, 1, 1, 2, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 2, 2, 2, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 1, 0, 2, 0, 1, 0, 2, 2, 0, 0, 0, 2, 2, 2, 1, 0, 0, 2, 0, 0, 0, 0, 1, 2, 2, 1, 2, 0, 2, 2, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 3, 0, 1, 0, 0, 2, 2, 0

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = q + q^2 + q^3 + q^4 + q^6 + 2*q^7 + q^8 + q^9 + q^12 + 2*q^13 + 2*q^14 + ...

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A093829, A093766, A097197, A107760, A112604, A112605.
Cf. A112606, A112607, A112608, A121361, A123884, A131961, A131962.
Cf. A131963, A131964, A137608.
Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(6), 1), 88); B := (A[1] - 1) / 3 + A[2]; B; /* Michael Somos, Aug 04 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* Michael Somos, Aug 04 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] - 4) / 12, {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
a[n_] := DivisorSum[n, KroneckerSymbol[-12, #]&]; Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d)))}; /* Michael Somos, Apr 18 2004 */

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X))) [n])}; /* Michael Somos, Jun 24 2011 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^2 + A)^6 / (eta(x^6 + A)^2 * eta(x + A)^3) - 1) / 3, n))}; /* Michael Somos, Aug 11 2009 */

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%6==5, 1-e%2, 1+e)))}; /* Michael Somos, Aug 04 2015 */

Moebius transform is period 6 sequence [ 1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Feb 14 2006
G.f. A(x) satisfies 0 = f(A(x),  A(x^2),  A(x^3),  A(x^6)) where f(u1, u2, u3, u6) = (u1 - u2) * (u1 - u2 - u3 + u6) - (u2 -u6) * (1 + 3*u6). - Michael Somos, May 29 2005
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -12, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -12, p) * p^-s)). - Michael Somos, Jun 24 2011
a(n) is multiplicative with a(p^e) = 1 if p=2 or p=3, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) = Sum_{k>=0} x^(6*k + 1) / (1 - x^(6*k + 1)) - x^(6*k + 5) / (1 - x^(6*k + 5)). - Michael Somos, Feb 14 2006
a(n) = |A093829(n)| = -(-1)^n * A137608(n) = a(2*n) = a(3*n).  a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
From Michael Somos, Aug 11 2009: (Start)
3 * a(n) = A107760(n) unless n=0. a(2*n + 1) = A033762(n). a(3*n + 1) = A033687(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 7) = 2 * A112608(n). a(12*n + 1) A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n). (End)
Expansion of (psi(q)^3 / psi(q^3) - 1) / 3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 16 2023

Definition edited by Michael Somos, Aug 11 2009

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

A222072 Decimal expansion of (1/384)*Pi^4.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 10 2013

Conjectured to be density of densest packing of equal spheres in 8 dimensions (achieved for example by the E_8 lattice).

The above conjecture is true (cf. Viazovska, 2017). - Felix Fröhlich, Jan 08 2018

			.25366950790104801363656336637683622721283225435595161898819...

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?, Discrete & Computational Geometry, Vol. 13, No. 3-4 (1995), 383-403.
G. Nebe and N. J. A. Sloane, Home page for E_8 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).
Maryna S. Viazovska, The sphere packing problem in dimension 8, Annals of Mathematics, Vol. 185, No. 3 (2017), 991-1015.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv:1603.04246 [math.NT], 2017.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093766, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A260646.

RealDigits[Pi^4/ 384,10,120][[1]] (* Harvey P. Dale, Aug 11 2015 *)

Pi^4/384 \\ Charles R Greathouse IV, Oct 31 2014

Equals Sum_{n>=1} Sum_{k>=n} 1/(2*n - 1)^2/(2*k + 1)^2. - Geoffrey Critzer, Nov 03 2013

A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

Original entry on oeis.org

1, 0, 1, 3, 0, 0, 2, 0, 1, 0, 0, 3, 2, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 6, 0, 0, 2, 0, 1, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0

Offset: 1

Michael Somos, Jul 15 2004

			G.f. = x + x^3 + 3*x^4 + 2*x^7 + x^9 + 3*x^12 + 2*x^13 + 3*x^16 + 2*x^19 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).

Antti Karttunen, Table of n, a(n) for n = 1..65537
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.

Cf. A033716, A093766, A115979.

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then  a := a+1 ; end if; end do: a; end proc:
A002324 := proc(n) sigmamr(n,3,1)-sigmamr(n,3,2) ; end proc:
A096936 := proc(n) A002324(n) +2*(sigmamr(n,12,4)-sigmamr(n,12,8) ); end proc:
seq(A096936(n),n=1..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1 || # == 3, 1, # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 1, #2 + 1, True, (1 + (-1)^#2)/2] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 20 2017 *)

{a(n) = if( n<1, 0, 1/2 * polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n)) * sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x*O(x^n)), n))};

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 3], n)[n])}; /* Michael Somos, Jun 05 2005 */

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, 3 * (1 + (-1)^e) / 2, p%3==2, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Nov 20 2017 */

(definec (A096936 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n)) (rest (A096936 (A028234 n)))) (cond ((= 2 p) (* (if (odd? e) 0 3) rest)) ((= 3 p) rest) ((= 1 (modulo p 3)) (* (+ 1 e) rest)) (else (* (if (odd? e) 0 1) rest)))))) ;; With the memoization-macro definec, after the given multiplicative formula. - Antti Karttunen, Nov 20 2017

a(n) = A033716(n) / 2.
Multiplicative with a(2^e) = 3*(1+(-1)^e)/2, a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if p==2 (mod 3) and a(p^e) = 1+e if p==1 (mod 3). - Corrected by Antti Karttunen, Nov 20 2017
G.f.: ((Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(3*k^2)) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Oct 15 2022

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