A346585 -id:A346585 - cp's OEIS frontend

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

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

Offset: 0

Eric W. Weisstein, Apr 15 2004

Density of densest packing of equal circles in two dimensions (achieved for example by the A2 lattice).
The number gives the areal coverage (90.68... percent) of the close hexagonal (densest) packing of circles in the plane. The hexagonal unit cell is a rhombus of side length 1 and height sqrt(3)/2; the area of the unit cell is sqrt(3)/2 and the four parts of circles add to an area of one circle of radius 1/2, which is Pi/4. - R. J. Mathar, Nov 22 2011
Ratio of surface area of a sphere to the regular octahedron whose edge equals the diameter of the sphere. - Omar E. Pol, Dec 09 2013

			0.906899682117108925297039128821077866142033124046370287784942...

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. 506.
L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (84) on page 16.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 30.

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.
Xi Lin, Dirk Schmelter, Sadaf Imanian, and Horst Hintze-Bruening, Hierarchically Ordered alpha-Zirconium Phosphate Platelets in Aqueous Phase with Empty Liquid, Scientific Reports (2019) Vol. 9, Article No. 16389.
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015. See Table 22 for L(m=6,r=2,s=1).
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Michael I. Shamos, A catalog of the real numbers, (2007). See pp. 625-626.
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).
Eckard Specht, May 21 2012, The best known packings of equal circles in a circle (complete up to N=1500).
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See p. 146.
Eric Weisstein's World of Mathematics, Smoothed Octagon.
Eric Weisstein's World of Mathematics, Circle Packing.
Index entries for transcendental numbers.

Related constants: A020769, A020789, A093825, A222066, A222067, A222068, A222069, A222070, A222071, A222072, A260646, (1/2)*A093602, A346585, A346584, A346583.

Cf. A072691, A381671.

RealDigits[Pi/(2 Sqrt[3]), 10, 111][[1]] (* Robert G. Wilson v, Nov 07 2012 *)

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

Equals (5/6)*(7/6)*(11/12)*(13/12)*(17/18)*(19/18)*(23/24)*(29/30)*(31/30)*..., where the numerators are primes > 3 and the denominators are the nearest multiples of 6.
Equals Sum_{n>=1} 1/A134667(n). [Jolley]
Equals Sum_{n>=0} (-1)^n/A124647(n). [Jolley eq. 273]
Equals A000796 / A010469. - Omar E. Pol, Dec 09 2013
Continued fraction expansion: 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). See A254381 for a sketch proof. - Peter Bala, Feb 04 2015
From Peter Bala, Feb 16 2015: (Start)
Equals 4*Sum_{n >= 0} 1/((6*n + 1)*(6*n + 5)).
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(2 + 7^2/(4 + 11^2/(2 + ... + (6*n + 1)^2/(4 + (6*n + 5)^2/(2 + ... ))))))). (End)
The inverse is (2*sqrt(3))/Pi = Product_{n >= 1} 1 + (1 - 1/(4*n))/(4*n*(9*n^2 - 9*n + 2)) = (35/32) * (1287/1280) * (8075/8064) * (5635/5632) * (72819/72800) * ... =  1.102657790843585... - Dimitris Valianatos, Aug 31 2019
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 3) dx.
Equals Integral_{x=0..oo} 1/(3*x^2 + 1) dx. (End)
Equals 1 + Sum_{k>=1} ( 1/(6*k+1) - 1/(6*k-1) ). - Sean A. Irvine, Jul 24 2021
For positive integer k, Pi/(2*sqrt(3)) = Sum_{n >= 0} (6*k + 4)/((6*n + 1)*(6*n + 6*k + 5)) - Sum_{n = 0..k-1} 1/(6*n + 5). - Peter Bala, Jul 10 2024
From Stefano Spezia, Jun 05 2025: (Start)
Equals Sum_{k>=0} (-1)^k/((k + 1)*(3*k + 1)).
Equals Integral_{x=0..oo} 1/(x^4 + x^2 + 1) dx.
Equals Integral_{x=0..oo} x^2/(x^4 + x^2 + 1) dx. (End)
Equals sqrt(A072691) = 3*A381671. - Hugo Pfoertner, Jun 05 2025

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

A122373 Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 4, 9, 16, 24, 36, 50, 64, 81, 96, 120, 144, 170, 200, 216, 256, 288, 324, 362, 384, 450, 480, 528, 576, 601, 680, 729, 800, 840, 864, 962, 1024, 1080, 1152, 1200, 1296, 1370, 1448, 1530, 1536, 1680, 1800, 1850, 1920, 1944, 2112, 2208, 2304, 2451, 2404

Offset: 1

Michael Somos, Aug 30 2006

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

a(n) = n^2 and n > 0 if and only if n = 2^i * 3^j with i, j >=0 (numbers in A003586). - Michael Somos, Jun 08 2012

			G.f. = q + 4*q^2 + 9*q^3 + 16*q^4 + 24*q^5 + 36*q^6 + 50*q^7 + 64*q^8 + 81*q^9 + 96*q^10 + ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Kevin Acres and David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See p. 3 eq. (3).
Mathew D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures, arXiv:0806.3590 [math.NT], 2008-2010. See p. 15, eq. (4.21).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A003586, A132000, A346585.

Cf. A000122, A000700, A010054, A121373.

terms = 50; QP = QPochhammer; s = QP[q^2]^5*QP[q^3]^4*(QP[q^6]/QP[q]^4) + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 04 2017, from first formula *)

{a(n) = my(A, p, e, f); if( n<0, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, p^(2*e), f =- (-1)^(p%3); (p^(2*e + 2) - f^(e+1)) / (p^2 - f))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^4 * eta(x^6 + A) / eta(x + A)^4, n))};

Expansion of eta(q^2)^5 * eta(q^3)^4 * eta(q^6) / eta(q)^4 in powers of q.
a(n) is multiplicative with a(2^e) = 4^e, a(3^e) = 9^e, a(p^e) = (p^(2*e + 2) - f^(e+1)) / (p^2 - f) where f = 1 if p == 1 (mod 6), f = -1 if p == 5 (mod 6).
Euler transform of period 6 sequence [4, -1, 0, -1, 4, -6, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 3^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is g.f. for A132000.
G.f.: Sum_{k>0} k^2 * x^k / (1 + x^k + x^(2*k)) * (1 + (1+(-1)^k)/8).
G.f.: Product_{k>0} (1 - x^k) * (1 + x^(3*k)) * (1 + x^k)^5 * (1 - x^(3*k))^5.
Expansion of psi(q)^2 * psi(q^3)^2 * phi(-q^3)^3 / phi(-q) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 23 2012
Expansion of c(q) * c(q^2) * b(q^2)^2 / (9 * b(q)) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 23 2012
G.f.: Sum_{k>0} k^2 * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)). - Michael Somos, Jul 05 2020
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/(18*sqrt(3)) = 0.994526... (A346585). - Amiram Eldar, Dec 22 2023

A346583 Decimal expansion of 301 * Pi^7 / (524880 * sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 24 2021

			0.999988377409405787862191368361863144410...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (314).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A093766, A346585, A346584, A346570.

First[RealDigits[N[7 * 43 * Pi^7 / (6! * 3^6 * Sqrt[3]),88]]] (* Stefano Spezia, Jul 24 2021 *)

Equals 7 * 43 * Pi^7 / (6! * 3^6 * sqrt(3)).

Equals 1 + Sum_{k>=1} ( 1/(6*k+1)^7 - 1/(6*k-1)^7 ).

A346584 Decimal expansion of 11 * Pi^5 / (1944 * sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 24 2021

			0.99973560764875173899502869478850690...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (314).

Michael I. Shamos, Shamos's catalog of the real numbers (2011).
Index entries for transcendental numbers

Cf. A093766, A346585, A346583, A346571.

First[RealDigits[N[11 * Pi^5 / (4! * 3^4 * Sqrt[3]),88]]] (* Stefano Spezia, Jul 24 2021 *)

Equals 11 * Pi^5 / (4! * 3^4 * sqrt(3)).

Equals 1 + Sum_{k>=1} ( 1/(6*k+1)^5 - 1/(6*k-1)^5 ).

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

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A122373 Expansion of (c(q)^3 + c(q^2)^3) / 27 in powers of q where c() is a cubic AGM theta function.

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A346583 Decimal expansion of 301 * Pi^7 / (524880 * sqrt(3)).

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A346584 Decimal expansion of 11 * Pi^5 / (1944 * sqrt(3)).

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