A186706 -id:A186706 - cp's OEIS frontend

A004016 Theta series of planar hexagonal lattice A_2.

1, 6, 0, 6, 6, 0, 0, 12, 0, 6, 0, 0, 6, 12, 0, 0, 6, 0, 0, 12, 0, 12, 0, 0, 0, 6, 0, 6, 12, 0, 0, 12, 0, 0, 0, 0, 6, 12, 0, 12, 0, 0, 0, 12, 0, 0, 0, 0, 6, 18, 0, 0, 12, 0, 0, 0, 0, 12, 0, 0, 0, 12, 0, 12, 6, 0, 0, 12, 0, 0, 0, 0, 0, 12, 0, 6, 12, 0, 0, 12, 0

Offset: 0

N. J. A. Sloane

The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
a(n) is the number of integer solutions to x^2 + x*y + y^2 = n (or equivalently x^2 - x*y + y^2 = n). - Michael Somos, Sep 20 2004
a(n) is the number of integer solutions to x^2 + y^2 + z^2 = 2*n where x + y + z = 0. - Michael Somos, Mar 12 2012
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (the present sequence), b(q) (A005928), c(q) (A005882).
a(n) = 6*A002324(n) if n>0, and A002324 is multiplicative, thus a(1)*a(m*n) = a(n)*a(m) if n>0, m>0 are relatively prime. - Michael Somos, Mar 17 2019
The first occurrence of a(n)= 6, 12, 18, 24, ... (multiples of 6) is at n= 1, 7, 49, 91, 2401, 637, 117649, ... (see A002324). - R. J. Mathar, Sep 21 2024

			G.f. = 1 + 6*x + 6*x^3 + 6*x^4 + 12*x^7 + 6*x^9 + 6*x^12 + 12*x^13 + 6*x^16 + ...
Theta series of A_2 on the standard scale in which the minimal norm is 2:
1 + 6*q^2 + 6*q^6 + 6*q^8 + 12*q^14 + 6*q^18 + 6*q^24 + 12*q^26 + 6*q^32 + 12*q^38 + 12*q^42 + 6*q^50 + 6*q^54 + 12*q^56 + 12*q^62 + 6*q^72 + 12*q^74 + 12*q^78 + 12*q^86 + 6*q^96 + 18*q^98 + 12*q^104 + 12*q^114 + 12*q^122 + 12*q^126 + 6*q^128 + 12*q^134 + 12*q^146 + 6*q^150 + 12*q^152 + 12*q^158 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 171, Entry 28.
Harvey Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Ex. 18.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
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..1000 from T. D. Noe)
S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338; F_3(q).
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
G. L. Hall, Comment on the paper "Theta series and magic numbers for diamond and certain ionic crystal structures" [J. Math. Phys. 28, 1653 (1987)]. Journal of Mathematical Physics; Sep. 1988, Vol. 29 Issue 9, pp. 2090-2092. - From _N. J. A. Sloane_, Dec 18 2012
M. D. Hirschhorn, Three classical results on representations of a number, Séminaire Lotharingien de Combinatoire, B42f (1999), 8 pp.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
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.]
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
Helmut Ruhland, A family of lattices with an unbounded number of unit vectors, arXiv:2410.16172 [math.MG], 2024. See p. 2.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
N. J. A. Sloane, Tables of Sphere Packings and Spherical Codes, IEEE Trans. Information Theory, vol. IT-27, 1981 pp. 327-338.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences related to A2 = hexagonal = triangular lattice.

Cf. A002324, A003051, A003215, A005881, A005882, A005928, A008458, A033685, A033687, A038587-A038591, etc.
See also A035019.
Cf. A000007, A000122, A004015, A008444, A008445, A008446, A008447, A008448, A008449 (Theta series of lattices A_0, A_1, A_3, A_4, ...), A186706.

Basis( ModularForms( Gamma1(3), 1), 81) [1]; /* Michael Somos, May 27 2014 */

L := Lattice("A",2); A := ThetaSeries(L, 161); A; /* Michael Somos, Nov 13 2014 */

A004016 := proc(n)
    local a,j ;
    a := A033716(n) ;
    for j from 0 to n/3 do
        a := a+A089800(n-1-3*j)*A089800(j) ;
    end do:
    a;
end proc:
seq(A004016(n),n=0..49) ; # R. J. Mathar, Feb 22 2021

a[ n_] := If[ n < 1, Boole[ n == 0 ], 6 DivisorSum[ n, KroneckerSymbol[ #, 3] &]]; (* Michael Somos, Nov 08 2011 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3], {q, 0, n}]; (* Michael Somos, Nov 13 2014 *)
a[ n_] := Length @ FindInstance[ x^2 + x y + y^2 == n, {x, y}, Integers, 10^9]; (* Michael Somos, Sep 14 2015 *)
terms = 81; f[q_] = LatticeData["A2", "ThetaSeriesFunction"][-I Log[q]/Pi]; s = Series[f[q], {q, 0, 2 terms}]; CoefficientList[s, q^2][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 6 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p%3==1, e+1, 1-e%2)))}; /* Michael Somos, May 20 2005 */ /* Editor's note: this is the most efficient program */

{a(n) = if( n<0, 0, polcoeff( 1 + 6 * sum( k=1,n, x^k / (1 + x^k + x^(2*k)), x * O(x^n)), n))}; /* Michael Somos, Oct 06 2003 */

{a(n) = if( n<1, n==0, 6 * sumdiv( n,d, kronecker( d, 3)))}; /* Michael Somos, Mar 16 2005 */

{a(n) = if( n<1, n==0, 6 * sumdiv( n,d, (d%3==1) - (d%3==2)))}; /* Michael Somos, May 20 2005 */

{a(n) = my(A); if( n<0, 0, n*=3; A = x * O(x^n); polcoeff( (eta(x + A)^3  + 3 * x * eta(x^9 + A)^3) / eta(x^3 + A), n))}; /* Michael Somos, May 20 2005 */

{a(n) = if( n<1, n==0, qfrep([ 2, 1; 1, 2], n, 1)[n] * 2)}; /* Michael Somos, Jul 16 2005 */

{a(n) = if( n<0, 0, polcoeff( 1 + 6 * sum( k=1, n, x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1) / (1 - x^(3*k - 1)), x * O(x^n)), n))} /* Paul D. Hanna, Jul 03 2011 */

from math import prod
from sympy import factorint
def A004016(n): return 6*prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) if n else 1 # Chai Wah Wu, Nov 17 2022

ModularForms( Gamma1(3), 1, prec=81).0 ; # Michael Somos, Jun 04 2013

Expansion of a(q) in powers of q where a(q) is the first cubic AGM theta function.
Expansion of theta_3(q) * theta_3(q^3) + theta_2(q) * theta_2(q^3) in powers of q.
Expansion of phi(x) * phi(x^3) + 4 * x * psi(x^2) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (1 / Pi) integral_{0 .. Pi/2} theta_3(z, q)^3 + theta_4(z, q)^3 dz in powers of q^2. - Michael Somos, Jan 01 2012
Expansion of coefficient of x^0 in f(x * q, q / x)^3 in powers of q^2 where f(,) is Ramanujan's general theta function. - Michael Somos, Jan 01 2012
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 3*v^2 - 2*u*w + 4*w^2. - Michael Somos, Jun 11 2004
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-u3) * (u3-u6) - (u2-u6)^2. - Michael Somos, May 20 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 11 2007
G.f. A(x) satisfies A(x) + A(-x) = 2 * A(x^4), from Ramanujan.
G.f.: 1 + 6 * Sum_{k>0} x^k / (1 + x^k + x^(2*k)). - Michael Somos, Oct 06 2003
G.f.: Sum_( q^(n^2+n*m+m^2) ) where the sum (for n and m) extends over the integers. - Joerg Arndt, Jul 20 2011
G.f.: theta_3(q) * theta_3(q^3) + theta_2(q) * theta_2(q^3) = (eta(q^(1/3))^3 + 3 * eta(q^3)^3) / eta(q).
G.f.: 1 + 6*Sum_{n>=1} x^(3*n-2)/(1-x^(3*n-2)) - x^(3*n-1)/(1-x^(3*n-1)). - Paul D. Hanna, Jul 03 2011
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = 6 * A033687(n). - Michael Somos, Jul 16 2005
a(2*n + 1) = 6 * A033762(n), a(4*n + 2) = 0, a(4*n) = a(n), a(4*n + 1) = 6 * A112604(n), a(4*n + 3) = 6 * A112595(n). - Michael Somos, May 17 2013
a(n) = 6 * A002324(n) if n>0. a(n) = A005928(3*n).
Euler transform of A192733. - Michael Somos, Mar 12 2012
a(n) = (-1)^n * A180318(n). - Michael Somos, Sep 14 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(3) = 3.627598... (A186706). - Amiram Eldar, Oct 15 2022

A073005 Decimal expansion of Gamma(1/3).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and exp(sqrt(3)*Pi) over Q. - Charles R Greathouse IV, Nov 11 2013

			Gamma(1/3) = 2.6789385347077476336556929409746776441286893779573011009...

H. B. Dwight, Tables of Integrals and other Mathematical Data. 860.18, 860.19 in Definite Integrals. New York, U.S.A.: Macmillan Publishing, 1961, p. 230.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Alessandro Languasco and Pieter Moree, Euler constants from primes in arithmetic progression, arXiv:2406.16547 [math.NT], 2024. See p. 8.
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics 187:9 (1996), pp. 1319-1348. (English translation)
Andrea Pinos, Gamma of reciprocal by Laplace.
Simon Plouffe, GAMMA(1/3).
Wikipedia, Particular values of the Gamma function: General rational arguments.
Index entries for transcendental numbers.
Index to sequences related to the Gamma function.

Cf. A073006, A202623, A203145, A256165.

R:= RealField(100); SetDefaultRealField(R); Gamma(1/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[1/3], 110]][[1]]
```

default(realprecision, 1080); x=gamma(1/3); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b073005.txt", n, " ", d)); \\ Harry J. Smith, Apr 19 2009

this * A073006 = A186706. - R. J. Mathar, Jan 15 2021
From Amiram Eldar, Jun 25 2021: (Start)
Equals 2^(7/9) * Pi^(1/3) * K((sqrt(3)-1)/(2*sqrt(2)))^(1/3)/3^(1/12), where K is the complete elliptic integral of the first kind.
Equals 2^(7/9) * Pi^(2/3) /(AGM(2, sqrt(2+sqrt(3)))^(1/3) * 3^(1/12)), where AGM is the arithmetic-geometric mean. (End)
From Andrea Pinos, Aug 12 2023: (Start)
Equals Integral_{x=0..oo} 3*exp(-(x^3)) dx = 3*A202623.
General result: Gamma(1/n) = Integral_{x=0..oo} n*exp(-(x^n)) dx. (End)
Equals 3*A202623 = exp(A256165). - Hugo Pfoertner, Jun 28 2024
Equals (2^(1/3)*Pi*C*3^(1/2))^(1/3), where C = A118292 = Integral {0..1} 2/sqrt(1-x^3) is the transcendental butterfly constant. - Jan Lügering, Feb 08 2025

A073006 Decimal expansion of Gamma(2/3).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

This constant is transcendental: Chudnovsky famously proved that Gamma(1/3) is algebraically independent of Pi, but Gamma(1/3)*Gamma(2/3) = 2*Pi/sqrt(3) by the reflection formula. - Charles R Greathouse IV, Aug 21 2023

			1.354117939426400416945288028154513785519327266056793698394022467963782...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Simon Plouffe, GAMMA(2/3).
Index to sequences related to the Gamma function.
Index entries for transcendental numbers.

Cf. A030652 (continued fraction). - Harry J. Smith, May 14 2009

Cf. A073005, A186706.

SetDefaultRealField(RealField(100)); Gamma(2/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[2/3], 110]][[1]]
```

allocatemem(932245000); default(realprecision, 5080); x=gamma(2/3); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b073006.txt", n, " ", d));  \\ Harry J. Smith, May 14 2009

Gamma(2/3) * A073005 = A186706. - R. J. Mathar, Jun 18 2006

A248897 Decimal expansion of Sum_{i >= 0} (i!)^2/(2*i+1)!.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 06 2015

Value of the Borwein-Borwein function I_3(a,b) for a = b = 1. - Stanislav Sykora, Apr 16 2015

The area of a circle circumscribing a unit-area regular hexagon. - Amiram Eldar, Nov 05 2020

			1.2091995761561452337293855050947704881893774987284937170465899569254...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), pp. 120-121.
L. B. W. Jolley, Summation of Series, Dover (1961), No. 261, pp. 48, 49, (and No. 275).

Xavier Gourdon and Pascal Sebah, Collection of series for Pi (see paragraph 7).
Su Hu and Min-Soo Kim, A generalization of Wallis' formula, arXiv:2201.09674 [math.NT], 2022.
Richard Kershner, The Number of Circles Covering a Set, American Journal of Mathematics, 61(3), 665-671.
MIT Integration Bee, 2023 MIT Integration Bee - Finals, Problem 1.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), C6.2.
Renzo Sprugnoli, Sums of reciprocals of the central binomial coefficients, INTEGERS 6 (2006) #A27
László Fejes Tóth, An Inequality concerning polyhedra, Bull. Amer. Math. Soc. 54 (1948), 139-146. See (9) p. 146.
Eric Weisstein's World of Mathematics, Arithmetic-Geometric Mean, equations 26-32.
Index entries for transcendental numbers

Cf. A000796, A001651, A002194, A093602, A248181, A257096, A257097, A186706.
Cf. A010815, A073010, A086089, A214549, A256923, A275486.
Cf. A091682 (Sum_{i >= 0} (i!)^2/(2*i)!).

RealDigits[2 Sqrt[3] Pi/9, 10, 100][[1]]

a = 2*Pi/(3*sqrt(3)) \\ Stanislav Sykora, Apr 16 2015

Equals 2*sqrt(3)*Pi/9 = 1 + 1/6 + 1/30 + 1/140 + 1/630 + 1/2772 + 1/12012 + ...
Equals m*I_3(m,m) = m*Integral_{x>=0} (x/(m^3+x^3)), for any m>0. - Stanislav Sykora, Apr 16 2015
Equals Integral_{x>=0} (1/(1+x^3)) dx. - Robert FERREOL, Dec 23 2016
From Peter Bala, Oct 27 2019: (Start)
Equals 3/4*Sum_{n >= 0} (n+1)!*(n+2)!/(2*n+3)!.
Equals Sum_{n >= 1} 3^(n-1)/(n*binomial(2*n,n)).
Equals 2*Sum_{n >= 1} 1/(n*binomial(2*n,n)). See Boros and Moll, pp. 120-121.
Equals Integral_{x = 0..1} 1/(1 - x^3)^(1/3) dx = Sum_{n >= 0} (-1)^n*binomial(-1/3,n) /(3*n + 1).
Equals 2*Sum_{n >= 1} 1/((3*n-1)*(3*n-2)) = 2*(1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...) (added Oct 30 2019). (End)
Equals Product_{k>=1} 9*k^2/(9*k^2 - 1). - Amiram Eldar, Aug 04 2020
From Peter Bala, Dec 13 2021: (Start)
Equals (2/3)*A093602.
Conjecture: for k >= 0, 2*sqrt(3)*Pi/9 = (3/2)^k * k!*Sum_{n = -oo..oo} (-1)^n/ Product_{j = 0..k} (3*n + 3*j + 1). (End)
Equals (3/4)*S - 1, where S = A248682. - Peter Luschny, Jul 22 2022
Equals Integral_{x=0..Pi/2} tan(x)^(1/3)/(sin(2*x) + 1) dx. See MIT Link. - Joost de Winter, Aug 26 2023
Continued fraction: 1/(1 - 1/(7 - 12/(12 - 30/(17 - ... - 2*n*(2*n - 1)/((5*n + 2) - ... ))))). See A000407. - Peter Bala, Feb 20 2024
Equals Sum_{n>=2} 1/binomial(n, floor(n/2)); and trivially if "floor" is replaced by "ceiling". - Richard R. Forberg, Aug 30 2024
Equals Product_{k>=2} (1 + (-1)^k/A001651(k)). - Amiram Eldar, Nov 22 2024
Equals 2*A073010 = 1/A086089 = sqrt(A214549) = exp(A256923) = A275486/2. - Hugo Pfoertner, Nov 22 2024
Equals 1 - (1/6) * Sum_{n>=1} A010815(n)/n. - Friedjof Tellkamp, Apr 05 2025
Equals A248181 - 2. - Pontus von Brömssen, Apr 05 2025

A247719 Decimal expansion of Integral_{t=0..Pi/2} sqrt(tan(t)) dt.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 23 2014

			2.22144146907918312350794049503034684930731...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey and J. M. Borwein, Highly Parallel, High-Precision Numerical Integration p. 7. (2005) Lawrence Berkeley National Laboratory.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, pp. 235-236.

Cf. A063448, A093954, A193887, A244976, A181048, A181049, A186706.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sqrt(2); // G. C. Greubel, Sep 07 2018

RealDigits[Pi/Sqrt[2], 10, 104] // First

default(realprecision, 100); Pi/sqrt(2) \\ G. C. Greubel, Sep 07 2018

Equals Pi/sqrt(2).
Equals A063448/2.
c = 2*( Sum_{k >= 0} (-1)^k/(4*k + 1) + Sum_{k >= 0} (-1)^k/(4*k + 3) ) = 2*(A181048 + A181049). - Peter Bala, Sep 21 2016
From Amiram Eldar, Aug 07 2020: (Start)
Equals Integral_{x=0..Pi} 1/(cos(x)^2 + 1) dx = Integral_{x=0..Pi} 1/(sin(x)^2 + 1) dx.
Equals Integral_{x=-oo..oo} 1/(x^4 + 1) dx.
Equals Integral_{x=-oo..oo} x^2/(x^4 + 1) dx.
Equals Integral_{x=0..oo} log(1 + 1/(2 * x^2)) dx. (End)
Equals Integral_{x=0..2*Pi} 1/(3 + 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 19 2023
Equals 20/9 - 160*Sum_{n >= 1} 1/((64*n^2 - 1)*(64*n^2 - 4)*(64*n^2 - 9)). - Peter Bala, Nov 09 2023

A053458 Open disk numbers (version 3): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (0,0).

Original entry on oeis.org

0, 1, 1, 7, 13, 19, 31, 43, 55, 73, 85, 109, 121, 151, 169, 199, 235, 253, 295, 313, 361, 397, 433, 475, 511, 571, 595, 661, 703, 757, 817, 859, 925, 979, 1039, 1111, 1159, 1237, 1285, 1381, 1453, 1519, 1597, 1663, 1759, 1813, 1915, 1993, 2077, 2173, 2257

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 13 2000

a(n)/(n/2)^2 --> Pi*2/sqrt(3) (see A186706).

Cf. A053416 (closed disk), A053456, A053457, A053459.

Cf. A186706.

A053458 := proc(d)
    local a,j,imin,imax ;
    a := 0 ;
    for j from -floor(d/sqrt(3)) do
        if j^2*3 >= d^2 and j>= 0 then
            break ;
        end if;
        imin := (-j-sqrt(d^2-3*j^2))/2 ;
        if type(imin,integer) then
            imin := imin+1 ;
        else
            imin := ceil(imin) ;
        end if;
        imax := (-j+sqrt(d^2-3*j^2))/2 ;
        if type(imax,integer) then
            imax := imax -1 ;
        else
            imax := floor(imax) ;
        end if;
        a := a+imax-imin+1 ;
    end do:
    a ;
end proc:
seq(A053458(d),d=0..30) ; # R. J. Mathar, Nov 22 2022

a[n_] := Sum[Boole[4*(i^2 + i*j + j^2) < n^2], {i, -n, n}, {j, -n, n}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 05 2023 *)

A244339 Expansion of (-2 * a(q) + 3a(q^2) + 2a(q^4)) / 3 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -4, 6, -4, 0, 0, 6, -8, 6, -4, 0, 0, 0, -8, 12, 0, 0, 0, 6, -8, 0, -8, 0, 0, 6, -4, 12, -4, 0, 0, 0, -8, 6, 0, 0, 0, 0, -8, 12, -8, 0, 0, 12, -8, 0, 0, 0, 0, 0, -12, 6, 0, 0, 0, 6, 0, 12, -8, 0, 0, 0, -8, 12, -8, 0, 0, 0, -8, 0, 0, 0, 0, 6, -8, 12, -4, 0

Offset: 0

Michael Somos, Jun 26 2014

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 4*q + 6*q^2 - 4*q^3 + 6*q^6 - 8*q^7 + 6*q^8 - 4*q^9 - 8*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A033762, A097195, A121373, A186706, A244375.

a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ (-1)^(n/d) {2, -1, 0, 1, -2, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^4 QPochhammer[ q^4] QPochhammer[ q^6]^8 / (QPochhammer[ q^2]^4 QPochhammer[ q^3]^4 QPochhammer[ q^12]^3), {q, 0, n}];

{a(n) = if( n<1, n==0, 2 * sumdiv(n, d, (-1)^(n/d) * [0, 2, -1, 0, 1, -2][d%6 + 1]))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A) * eta(x^6 + A)^8 / (eta(x^2 + A)^4 * eta(x^3 + A)^4 * eta(x^12 + A)^3), n))};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^k) * [0, -2, 1, 0, -1, 2][k%6 + 1], x * O(x^n)), n))};

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [3, -2, 1, -2][k%4 + 1], x * O(x^n)), n))};

{a(n) = my(A);  if( n<1, n==0, A = factor(n); -4 * prod( j=1, matsize(A)[1], if( p = A[j,1], e = A[j,2]; if( p==2, (1 - (-1)^e) * -3/4, if( p==3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2))))))};

Expansion of b(q) * (b(q) + 2*b(q^4)) / (3 * b(q^2)) in powers of q where b() is a cubic AGM theta function.
Expansion of psi(-q) * chi(-q)^3 * phi(q^3) * chi(q^3)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q)^4 * eta(q^4) * eta(q^6)^8 / (eta(q^2)^4 * eta(q^3)^4 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -4, 0, 0, -1, -4, -4, -4, -1, 0, 0, -4, -2, ...].
Moebius transform is period 12 sequence [ -4, 10, 0, -6, 4, 0, -4, 6, 0, -10, 4, 0, ...].
a(n) = -4 * b(n) where b(n) is multiplicative with b(2^e) = (1 - (-1)^e) * -3/4 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6) with a(0) = 1.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 48^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244375.
a(2*n) = A004016(n). a(2*n + 1) = -4 * A033762(n). a(3*n) = a(n). a(6*n + 1) = -4 * A097195(n). a(6*n + 5) = 0.
Sum_{k=1..n} abs(a(k)) ~ (2*Pi/sqrt(3)) * n. - Amiram Eldar, Jun 08 2025

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A053458 Open disk numbers (version 3): a(n) is the number of points (i+j/2,j*sqrt(3)/2), i,j integers (triangular grid) contained in an open disk of diameter n, centered at (0,0).

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