A005882 -id:A005882 - 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

A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

Original entry on oeis.org

1, -3, 0, 6, -3, 0, 0, -6, 0, 6, 0, 0, 6, -6, 0, 0, -3, 0, 0, -6, 0, 12, 0, 0, 0, -3, 0, 6, -6, 0, 0, -6, 0, 0, 0, 0, 6, -6, 0, 12, 0, 0, 0, -6, 0, 0, 0, 0, 6, -9, 0, 0, -6, 0, 0, 0, 0, 12, 0, 0, 0, -6, 0, 12, -3, 0, 0, -6, 0, 0, 0, 0, 0, -6, 0, 6, -6, 0, 0, -6, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, -12, 0, 12, 0, 0, 0, -6, 0, 0

Offset: 0

N. J. A. Sloane

Unsigned sequence is expansion of theta series of hexagonal net with respect to a node.
Cubic AGM theta functions: a(q) (see A004016), b(q) (this: A005928), c(q) (A005882).
Denoted by a_3(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - 3*q + 6*q^3 - 3*q^4 - 6*q^7 + 6*q^9 + 6*q^12 - 6*q^13 - 3*q^16 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.34).
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)
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.
J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47.
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. [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 below.]
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.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A005882, A033762, A097195, A113062, A123477, A123530, A246838, A253243.

A := Basis( ModularForms( Gamma1(9), 1), 100); A[1] - 3*A[2] + 6*A[4]; // Michael Somos, Jan 31 2015

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := If[ n < 1, Boole[ n==0], -3 Sum[{1, -1, -3, 1, -1, 3, 1, -1, 0}[[ Mod[ d, 9, 1]]], {d, Divisors @ n}]]; (* Michael Somos, Sep 23 2013 *)

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, if( p%6==1, e+1, !(e%2)))))}; \\ Michael Somos, May 20 2005

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

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, -3, 3, 9, -3, 3, -9, -3, 3] [d%9 + 1]))}; \\ Michael Somos, Dec 25 2007

N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(n)-sigma(3*n))*x^n/n));
Vec(gf) \\ Joerg Arndt, Jul 30 2011

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3/eta(q^3))} \\ Altug Alkan, Mar 20 2018

a(n) is the coefficient of q^n in b(q)=eta(q)^3/eta(q^3) = (3/2)*a(q^3)-a(q)/2 where a(q)=theta(Hexagonal). - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
From Michael Somos, May 20 2005: (Start)
Euler transform of period 3 sequence [ -3, -3, -2, ...].
a(n) = -3 * b(n) except for a(0) = 1, where b()=A123477() is multiplicative with b(p^e) = -2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 2, 5 (mod 6).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - 2*u*w^2 + u^2*w.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2*u6 - 2*u1*u2*u6 + 4*u2^2*u6 - 3*u2*u3^2.
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*u3 + u1^2*u3 - 3*u1*u6^2 + u2^2*u3. (End)
a(3*n + 2) = 0. a(3*n + 1) = -A005882(n), a(3*n) = A004016(n). - Michael Somos, Jul 15 2005
a(n) = -3 * A123477(n) unless n=0. |a(n)| = A113062(n).
Moebius transform is period 9 sequence [-3, 3, 9, -3, 3, -9, -3, 3, 0, ...]. - Michael Somos, Dec 25 2007
Expansion of b(q) = a(q^3) - c(q^3) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033687.
G.f.: exp( Sum_{n>=1} (sigma(n)-sigma(3*n))*x^n/n ). - Joerg Arndt, Jul 30 2011
a(n) = (-1)^(mod(n, 3) = 1) * A113062(n). - Michael Somos, Sep 05 2014
a(2*n + 1) = -3 * A123530(n). a(4*n) = a(n). a(4*n + 1) = -3 * A253243(n). a(4*n + 2) = 0. a(4*n + 3) = 6 * A246838(n). a(6*n + 1) = -3 * A097195(n). a(6*n + 3) = 6 * A033762(n). - Michael Somos, Jun 04 2015
G.f.: 1 + Sum_{k>0} -3 * x^k / (1 + x^k + x^(2*k)) + 9 * x^(3*k) / (1 + x^(3*k) + x^(6*k)). - Michael Somos, Jun 04 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

Edited by M. F. Hasler, May 07 2018

A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

Original entry on oeis.org

1, 240, 2160, 6720, 17520, 30240, 60480, 82560, 140400, 181680, 272160, 319680, 490560, 527520, 743040, 846720, 1123440, 1179360, 1635120, 1646400, 2207520, 2311680, 2877120, 2920320, 3931200, 3780240, 4747680, 4905600, 6026880

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
E_8 is also the Barnes-Wall lattice in 8 dimensions.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
The E_8 lattice is integral, unimodular, and even. The 240 shortest nonzero vectors in the lattice have norm squared 2. Of these vectors, 128 are all half-integer, and 112 are all integer. - Michael Somos, Jun 10 2019

			G.f. = 1 + 240*x + 2160*x^2 + 6720*x^3 + 17520*x^4 + 30240*x^5 + 60480*x^6 + ...
G.f. = 1 + 240*q^2 + 2160*q^4 + 6720*q^6 + 17520*q^8 + 30240*q^10 + 60480*q^12 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 123.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 53.
R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962, p. 53.
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, 1984, see p. 111.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Collected Papers of Srinivasa Ramanujan, Chap. 16, Ed. G. H. Hardy et al., Chelsea, NY, 1962.
S. Ramanujan, On Certain Arithmetical Functions, Messenger Math., 45 (1916), 11-15 (Eq. (25)). Ramanujan's Papers, p. 196, Ed. B. J. Venkatachala et al., Prism Books, Bangalore 2000.
Jean-Pierre Serre, "A Course in Arithmetic", Springer, 1978
Joseph H. Silverman, "Advanced Topics in the Arithmetic of Elliptic Curves", Springer, 1994
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
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 N. J. A. Sloane)
D. Bump, Automorphic Forms and Representations, Cambr. Univ. Press, 1997, p. 29.
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.
Heng Huat Chan, Shaun Cooper, and Pee Choon Toh, Ramanujan's Eisenstein series and powers of Dedekind's eta-function, Journal of the London Mathematical Society 75.1 (2007): 225-242. See Q(q).
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016.
H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561-578; reprinted in "Twelve Geometric Essays", pp. 20-39.
D. de Laat and F. Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
M. Kaneko and D. Zagier, Supersingular j-invariants, hypergeometric series and Atkin's orthogonal polynomials, pp. 97-126 of D. A. Buell and J. T. Teitelbaum, eds., Computational Perspectives on Number Theory, Amer. Math. Soc., 1998
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.]
Robert V. Moody and Jiri Patera, Fast recursion formula for weight multiplicities, Bulletin of the American Mathematical Society 7.1 (1982): 237-242.
G. Nebe and N. J. A. Sloane, Home page for E_8 lattice
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155.
Michael Somos, Introduction to Ramanujan theta functions
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.
Martin H. Weissman, Octonions, Cubes, Embeddings, March 2, 2009.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Eisenstein Series.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Wikipedia, Eisenstein series
Wikipedia, E_8 lattice
Index entries for sequences related to Eisenstein series
Index entries for sequences related to Barnes-Wall lattices

Cf. A046948 (partial sums), A000143, A108091 (eighth root).
Cf. A008410, A008411, A001158.
Cf. A006352 (E_2), A013973 (E_6), A008410 (E_8), A013974 (E_10), A029828 (E_12), A058550 (E_14), A029829 (E_16), A029830 (E_20), A029831 (E_24).
Cf. A007331 (theta_2(q)^8 / 256), A000143 (theta_3(q)^8), A035016 (theta_4(q)^8).

Basis( ModularForms( Gamma1(1), 4), 29) [1]; /* Michael Somos, May 11 2015 */

L := Lattice("E",8); A := ThetaSeries(L, 57); A; /* Michael Somos, Jun 10 2019 */

with(numtheory); E := proc(k) local n,t1; t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n,n=1..60); series(t1,q,60); end; E(4);

a[ n_] := If[ n < 1, Boole[n == 0], 240 DivisorSigma[ 3, n]]; (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 + 14 t2 t3 + t3^2], {q, 0, n}]; (* Michael Somos, Jun 04 2014 *)
max = 30; s = 1 + 240*Sum[k^3*(q^k/(1 - q^k)), {k, 1, max}] + O[q]^max; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, after Gene Ward Smith *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, t2^2 - t2 t3 + t3^2], {q, 0, 2 n}]; (* Michael Somos, Jul 31 2016 *)

{a(n) = if( n<1, n==0, 240 * sigma(n, 3))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^24 + 256 * x * eta(x^2 + A)^24) / (eta(x + A) * eta(x^2 + A))^8, n))}; /* Michael Somos, Dec 30 2008 */

q='q+O('q^50); Vec((eta(q)^24+256*q*eta(q^2)^24)/(eta(q)*eta(q^2))^8) \\ Altug Alkan, Sep 30 2018

from sympy import divisor_sigma
def a(n): return 1 if n == 0 else 240 * divisor_sigma(n, 3)
[a(n) for n in range(51)]  # Indranil Ghosh, Jul 15 2017

ModularForms(Gamma1(1), 4, prec=30).0 ; # Michael Somos, Jun 04 2013

Can also be expressed as E4(q) = 1 + 240*Sum_{i >= 1} i^3 q^i/(1 - q^i) - Gene Ward Smith, Aug 22 2006
Theta series of E_8 lattice = 1 + 240 * Sum_{m >= 1} sigma_3(m) * q^(2*m), where sigma_3(m) is the sum of the cubes of the divisors of m (A001158).
Expansion of (phi(-q)^8 - (2 * phi(-q) * phi(q))^4 + 16 * phi(q)^8) in powers of q where phi() is a Ramanujan theta function.
Expansion of (eta(q)^24 + 256 * eta(q^2)^24) / (eta(q) * eta(q^2))^8 in powers of q. - Michael Somos, Dec 30 2008
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 33*v^2 + 256*w^2 - 18*u*v + 16*u*w - 288*v*w . - Michael Somos, Jan 05 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) = u1^2 + 16*u2^2 + 81*u3^2 + 1296*u6^2 - 14*u1*u2 - 18*u1*u3 + 30*u1*u6 + 30*u2*u3 - 288*u2*u6 - 1134*u3*u6 . - Michael Somos, Apr 15 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^3), A(x^9)) where f(u, v, w) = u^3*v + 9*w*u^3 - 84*u^2*v^2 + 246*u*v^3 - 253*v^4 - 675*w*u^2*v + 729*w^2*u^2 - 4590*w*u*v^2 + 19926*w*v^3 - 54675*w^2*u*v + 59049*w^3*u + 531441*w^3*v - 551124*w^2*v^2 . - Michael Somos, Apr 15 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / t) = (t/i)^4 * f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 30 2008
Convolution square is A008410. A008411 is convolution of this sequence with A008410.
Expansion of Ramanujan's function Q(q^2) = 12 (omega/Pi)^4 g2 (Weierstrass invariant) in powers of q^2.
Expansion of a(q) * (a(q)^3 + 8*c(q)^3) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, Jan 14 2015
G.f. is (theta_2(q)^8 + theta_3(q)^8 + theta_4(q)^8) / 2 where q = exp(Pi i t). So a(n) = A008430(n) + 128*A007331(n) (= A000143(2*n) + 128*A007331(n) = A035016(2*n) + 128*A007331(n)). - Seiichi Manyama, Sep 30 2018
a(n) = 240*A001158(n) if n>0. - Michael Somos, Oct 01 2018
Sum_{k=1..n} a(k) ~ 2 * Pi^4 * n^4 / 3. - Vaclav Kotesovec, Jan 14 2024

A030203 Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.

Original entry on oeis.org

1, -1, -1, -1, 1, 2, -1, 2, 0, 0, -1, -1, -1, -1, 0, 1, -1, -1, 2, 0, 1, 2, 1, -1, 0, -1, 2, -1, 0, -1, -1, 0, -1, -1, 0, -1, -2, 2, 2, 0, -1, 1, 0, 1, 0, -1, 2, 2, 1, 0, -2, 2, -1, 0, -1, -1, -1, 1, -1, 0, 0, -1, -1, -1, 0, 0, 2, -2, -1, 0, -1, 1, 2, 2, 0, 0

Offset: 0

N. J. A. Sloane

sign

Number 65 of the 74 eta-quotients listed in Table I of Martin (1996).
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 - x - x^2 - x^3 + x^4 + 2*x^5 - x^6 + 2*x^7 - x^10 - x^11 - x^12 - x^13 + ...
G.f. = q - q^7 - q^13 - q^19 + q^25 + 2*q^31 - q^37 + 2*q^43 - q^61 - q^67 - ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. R. Finch, Powers of Euler's q-Series, (arXiv:math.NT/0701251).
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Basis( CuspForms( Gamma1(108), 1), 452)[1]; /* Michael Somos, Jan 31 2015 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = my(A, p, e); if( n<0, 0, n = 6*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%6==5, (1 + (-1)^e) / 2, (p-1) / znorder( Mod(2, p))%3, kronecker( e+1, 3), e+1)))}; /* Michael Somos, Jul 27 2006 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A), n))}; /* Michael Somos, Jul 27 2006 */

Expansion of f(-x) * f(-x^3) where f(-x) := f(-x, -x^2) is a Ramanujan theta function. - Michael Somos, Jul 27 2006
Expansion of q^(-1/6) * sqrt(b(q) * c(q)/3) in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 01 2006
Euler transform of period 3 sequence [ -1, -1, -2, ...]. - Michael Somos, Jul 27 2006
Given g.f. A(x), then B(q) = (q * A(q^6))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 - u^2*w - 4*u*w^2. - Michael Somos, Jul 27 2006
a(n) = b(6*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6), b(p^e) = e+1 if p = x^2 + 27*y^2, b(p^e) = [1, -1, 0] depending on e (mod 3) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (108 t)) = 108^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jan 22 2012
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)).

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

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)=0 if and only if A000731(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Number of 3-core partitions of n (denoted c_3(n) in Granville and Ono, p. 340). - Brian Hopkins, May 13 2008
Denoted by g_1(q) in Cynk and Hulek in Remark 3.4 on page 12 (but not explicitly listed).
This is a member of an infinite family of integer weight modular forms. g_1 = A033687, g_2 = A030206, g_3 = A130539, g_4 = A000731. - Michael Somos, Aug 24 2012
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + x + 2*x^2 + 2*x^4 + x^5 + 2*x^6 + x^8 + 2*x^9 + 2*x^10 + 2*x^12 + 2*x^14 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + 2*q^31 + 2*q^37 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.35) and (32.351).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Johnathan M. Borwein and Peter 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 697.
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Andrew Granville and Ken Ono, Defect Zero p-blocks for Finite Simple Groups, Transactions of the American Mathematical Society, Vol. 348 (1996), pp. 331-347.
Guo-Niu Han and Ken Ono, Hook Lengths and 3-Cores, Ann. Comb. (2011) Vol. 15, 305-312. See also arXiv:0805.2461, [math.NT], 2008.
Jeremy Lovejoy and Olivier Mallet, n-color overpartitions, twisted divisor functions, and Rogers-Ramanujan identities, South East Asian J. Math. Math. Sci., 6 (2008), 23-36. [From _Jeremy Lovejoy_, Jun 12 2009]
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000731, A002324, A005882, A006938, A033685.

Cf. A045831, A053723, A081622.

Basis( ModularForms( Gamma1(9), 1), 316) [2]; /* Michael Somos, May 06 2015 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Sep 23 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Sep 01 2015 *)

{a(n) = if( n<0, 0, sumdiv( 3*n + 1, d, kronecker( -3, d)))}; /* Michael Somos, Nov 03 2005 */

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor( 3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%6==1, e+1, 1-e%2)))}; /* Michael Somos, May 06 2015 */

Euler transform of period 3 sequence [1, 1, -2, ...].
Expansion of q^(-1/3) * eta(q^3)^3 / eta(q) in powers of q.
a(4*n + 1) = a(n). - Michael Somos, Dec 06 2004
a(n) = b(3*n + 1) where b(n) is multiplicative and b(p^e) = 0^e if p = 3, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 2, 5 (mod 6). - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - 2*u*w^2 - v^3. - Michael Somos, Dec 06 2004
Given g.f. A(x), B(q)= q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u3^2 + u1*u6^2 - u1*u3*u6 - u2^2*u3. - Michael Somos, May 20 2005
Given g.f. A(x), B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u2*u3^2 + 2*u2*u3*u6 + 4*u2*u6^2 - u1^2*u6. - Michael Somos, May 20 2005
G.f.: Product_{k>0} (1 - x^(3*k))^3 / (1 - x^k).
G.f.: Sum_{k in Z} x^k / (1 - x^(3*k + 1)) = Sum_{k in Z} x^k / (1 - x^(6*k + 2)). - Michael Somos, Nov 03 2005
Expansion of q^(-1) * c(q^3) / 3 = q^(-1) * (a(q) - b(q)) / 9 in powers of q^3 where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (27 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A005928.
a(n) = Sum_{d|3n+1} LegendreSymbol{d,3} - Brian Hopkins, May 13 2008
q-series for a(n): Sum_{n >= 0} q^(n^2+n)(1-q)(1-q^2)...(1-q^n)/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n+1))). [From Jeremy Lovejoy, Jun 12 2009]
a(n) = A002324(3*n + 1). 3*a(n) = A005882(n) = A033685(3*n + 1). - Michael Somos, Apr 04 2003
G.f.: (2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5)) / 3 where phi(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - Michael Somos, Sep 07 2018
Sum_{k=1..n} a(k) ~ 2*Pi*n/3^(3/2). - Vaclav Kotesovec, Dec 17 2022

A008620 Positive integers repeated three times.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26

Offset: 0

N. J. A. Sloane

Arises from Gleason's theorem on self-dual codes: the Molien series is 1/((1-x^8)*(1-x^24)) for the weight enumerators of doubly-even binary self-dual codes; also 1/((1-x^4)*(1-x^12)) for ternary self-dual codes.
The number of partitions of n into distinct parts where each part is either a power of two or three times a power of two.
Number of partitions of n into parts 1 or 3. - Reinhard Zumkeller, Aug 15 2011
The dimension of the space of modular forms on Gamma_1(3) of weight n>0 with a(q) the generator of weight 1 and c(q)^3 / 27 the generator of weight 3 where a(), c() are cubic AGM theta functions. - Michael Somos, Apr 01 2015
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
a(n-1) is the minimal number of circles that can be drawn through n points in general position in the plane. - Anton Zakharov, Dec 31 2016
Number of partitions of n into distinct parts from A029744.- R. J. Mathar, Mar 01 2023
Number of representations n=sum_i c_i*2^i with c_i in {0,1,3,4} [Anders]. See A120562 or A309025 for other c_i sets. - R. J. Mathar, Mar 01 2023

G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. page 12 Exer. 7
D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Eq. (14), p. 601 and Theorem 3c, p. 602; also Problem 5 p. 620.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 1.
E. R. Berlekamp, F. J. MacWilliams and N. J. A. Sloane, Gleason's Theorem on Self-Dual Codes, IEEE Trans. Information Theory, IT-18 (1972), 409-414.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 210
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 449
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Jan Snellman and Michael Paulsen, Enumeration of Concave Integer Partitions, J. Integer Seqs., Vol. 7, 2004.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A000027, A001840, A002264, A004016, A004396, A005882, A005928, A008621, A010766, A049347.

Column 3 of A235791.

a008620 = (+ 1) . (`div` 3)
a008620_list = concatMap (replicate 3) [1..]
-- Reinhard Zumkeller, Feb 19 2013, Apr 16 2012, Sep 25 2011

[Floor(n/3)+1: n in [0..80]]; // Vincenzo Librandi, Aug 16 2011

a := func< n | Dimension( ModularForms( Gamma1(3), n))>; /* Michael Somos, Apr 01 2015 */

A008620:=n->floor(n/3)+1; seq(A008620(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Table[Floor[n/3] + 1, {n, 0, 90}] (* Stefan Steinerberger, Apr 02 2006 *)
Table[{n, n, n}, {n, 30}] // Flatten (* Harvey P. Dale, Jan 15 2017 *)
Ceiling[Range[20]/3] (* Eric W. Weisstein, Aug 12 2023 *)
Table[Ceiling[n/3], {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(1 + n - Cos[2 n Pi/3] + Sin[2 n Pi/3]/Sqrt[3])/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
Table[(n - ChebyshevU[n, -1/2] + 1)/3, {n, 20}] (* Eric W. Weisstein, Aug 12 2023 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 1, 1, 2}, 20] (* Eric W. Weisstein, Aug 12 2023 *)
CoefficientList[Series[1/((-1 + x)^2 (1 + x + x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 12 2023 *)

PARI
```
a(n)=n\3+1
```

def a(n) : return( dimension_modular_forms( Gamma1(3), n) ); # Michael Somos, Apr 01 2015

a(n) = floor(n/3) + 1.
a(n) = A010766(n+3, 3).
G.f.: 1/((1-x)*(1-x^3)) = 1/((1-x)^2*(1+x+x^2)).
a(n) = A001840(n+1) - A001840(n). - Reinhard Zumkeller, Aug 01 2002
From Paul Barry, May 19 2004: (Start)
Convolution of A049347 and A000027.
a(n) = Sum_{k=0..n} (k+1)*2*sqrt(3)*cos(2*Pi*(n-k)/3 + Pi/6)/3. (End)
The g.f. is 1/(1-V_trefoil(x)), where V_trefoil is the Jones polynomial of the trefoil knot. - Paul Barry, Oct 08 2004
a(2n) = A004396(n+1). - Philippe Deléham, Dec 14 2006
a(n) = ceiling(n/3), n>=1. - Mohammad K. Azarian, May 22 2007
E.g.f.: exp(x)*(2 + x)/3 + exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 17 2022

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

nonn

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
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 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael D. Hirschhorn, Three classical results on representations of a number, Sem. Lotharingien de Combinat. S42 (1999), B42f.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions, 2010.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
From Michael Somos, Sep 18 2004: (Start)
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).
6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Corrected by Charles R Greathouse IV, Sep 02 2009

A214262 Expansion of eta(q)^5 * eta(q^3) * eta(q^6)^4 / eta(q^2)^4 in powers of q.

Original entry on oeis.org

1, -5, 9, -11, 24, -45, 50, -53, 81, -120, 120, -99, 170, -250, 216, -203, 288, -405, 362, -264, 450, -600, 528, -477, 601, -850, 729, -550, 840, -1080, 962, -821, 1080, -1440, 1200, -891, 1370, -1810, 1530, -1272, 1680, -2250, 1850, -1320, 1944, -2640, 2208

Offset: 1

Michael Somos, Jul 09 2012

Zagier (2009) writes "... associated to the weight 3 Eisenstein series g(z) = Sigma b(n)q^n = q - 5q^2 + 9q^3 - 11q^4 + ...".

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

			G.f. = q - 5*q^2 + 9*q^3 - 11*q^4 + 24*q^5 - 45*q^6 + 50*q^7 - 53*q^8 + 81*q^9 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Indranil Ghosh, Table of n, a(n) for n = 1..2000
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A111661.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# #^2 JacobiSymbol[ -3, n/#] &]]; (* Michael Somos, Oct 06 2013 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q]^5 QPochhammer[ q^3] QPochhammer[ q^6]^4 / QPochhammer[ q^2]^4, {q, 0, n}]; (* Michael Somos, Oct 06 2013 *)

{a(n) = if( n<1, 0, sumdiv( n, d, -(-1)^d * d^2 * kronecker( -3, n/d)))}; /* Michael Somos, Oct 06 2013 */

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor( n); prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==3, 9^e, if( p==2, -(4^(e+1) + 9*(-1)^(e+1)) / 5, if( p%6==1, ((p^2)^(e+1) - 1) / (p^2 - 1), ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1)))))))};

Expansion of (1/9) * c(q) * b(q)^2 * c(q^2) / b(q^2) = (c(q)^3 - 8*c(q^2)^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.
Euler transform of period 6 sequence [ -5, -1, -6, -1, -5, -6, ...]. - Michael Somos, Oct 06 2013
a(n) is multiplicative with a(3^e) = 9^e, a(2^e)  = -(4^(e+1) + 9*(-1)^(e+1)) / 5, a(p^e)  = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 6), a(p^e)  = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 192^(1/2) (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A111661.
G.f.: Sum_{k>0} -(-1)^k * k^2 * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker( -3, k) * (x^k - x^(2*k)) / (1 + x^k)^3.

A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18).

Original entry on oeis.org

1, -1, 0, 2, -2, -1, 4, -4, -1, 8, -8, -2, 14, -14, -4, 24, -23, -6, 40, -38, -10, 63, -60, -16, 98, -92, -24, 150, -140, -36, 224, -208, -54, 329, -304, -78, 478, -440, -112, 684, -627, -160, 968, -884, -224, 1358, -1236, -312, 1884, -1710, -432, 2592, -2346, -590, 3540, -3196, -801, 4796, -4320, -1082, 6454

Offset: 0

Michael Somos, Mar 07 2004

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 - x + 2*x^3 - 2*x^4 - x^5 + 4*x^6 - 4*x^7 - x^8 + 8*x^9 + ...
G.f. = q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A062242, A128111, A141094, A216046.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 - x^k, {k, 3, n, 6}]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)
a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}; /* Michael Somos, Dec 07 2013 */

{a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + (x*A^2)^2) - x*A^2); polcoeff(A, n))};

{a(n) = if( n<0, 0, polcoeff( prod(k=0, (n-1)\2, (1 - x^(2*k + 1))^if(k%3==1, -2, 1), 1 + x * O(x^n)), n))};

Expansion of chi(-q) / chi(-q^3)^3 where chi() is a Ramanujan theta function.
Expansion of q^(-1/3) * c(q^2) / c(q) where c() is a cubic AGM theta function. - Michael Somos, Oct 04 2006
Expansion of q^(-1/3) * eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [-1, 0, 2, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + 2*u*v^2.
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - v^2 + v) - u^3 * (1 + 2*v + 4*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A141094. - Michael Somos, Dec 07 2013
G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3.
G.f.: 1 / (1 + (x + x^2) / (1 + (x^2 + x^4) / (1 + (x^3 + x^6) / ...))).
a(n) = A062242(2*n + 1) = (-1)^n * A128111(n). Convolution inverse of A062242.
a(2*n + 1) = - A216046(n). Convolution square is A216046. - Michael Somos, Dec 07 2013
G.f.: T(0), where T(k) = 1 - (x^(k+1)+x^(2*k+2))/((x^(k+1)+x^(2*k+2))+1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013

A004011 Theta series of D_4 lattice; Fourier coefficients of Eisenstein series E_{gamma,2}.

Original entry on oeis.org

1, 24, 24, 96, 24, 144, 96, 192, 24, 312, 144, 288, 96, 336, 192, 576, 24, 432, 312, 480, 144, 768, 288, 576, 96, 744, 336, 960, 192, 720, 576, 768, 24, 1152, 432, 1152, 312, 912, 480, 1344, 144, 1008, 768, 1056, 288, 1872, 576, 1152, 96, 1368, 744, 1728, 336

Offset: 0

N. J. A. Sloane

D_4 is also the Barnes-Wall lattice in 4 dimensions.
E_{gamma,2} is the unique normalized modular form for Gamma_0(2) of weight 2.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).
Convolution square is A008658. - Michael Somos, Aug 21 2014
Expansion of 2*P(x^2) - P(x) in powers of x where P() is a Ramanujan Eisenstein series. - Michael Somos, Feb 16 2015
a(n) is the number of Hurwitz quaternions of norm n. - Michael Somos, Feb 16 2015

			G.f. = 1 + 24*x + 24*x^2 + 96*x^3 + 24*x^4 + 144*x^5 + 96*x^6 + 192*x^7 + 24*x^8 + ...
G.f. = 1 + 24*q^2 + 24*q^4 + 96*q^6 + 24*q^8 + 144*q^10 + 96*q^12 + 192*q^14 + 24*q^16 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1998, see p. 148 Eq. (9.11).
Harvey Cohn, Advanced Number Theory, Dover Publications, Inc., 1980, p. 89. Eq. (1).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 214.
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..10000
Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).
Barry Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, Vol. 7, No. 3 (1998), 257-274.
Michael Gilleland, Some Self-Similar Integer Sequences.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
Nadia Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
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.]
Gabriele Nebe and N. J. A. Sloane, Home page for D_4 lattice
N. J. A. Sloane, The 24 minimal vectors form the 24-cell polytope.
N. J. A. Sloane, Seven Staggering Sequences.
Eric Weisstein's World of Mathematics, 24-Cell.
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice.
Eric Weisstein's World of Mathematics, Eisenstein Series.
Wikipedia, Hurwitz quaternion.
Index entries for "core" sequences.
Index entries for sequences related to Barnes-Wall lattices.
Index entries for sequences related to D_4 lattice.
Index entries for sequences related to Eisenstein series.

Cf. A002388, A000118, A000593, A008658, A096727, A108092, A108096.
Cf. A004016, A005882, A005928.
Partial sums give A046949.
Cf. A108092 (convolution fourth root).

Basis( ModularForms( Gamma0(2), 2), 54) [1]; /* Michael Somos, May 27 2014 */

readlib(ifactors): with(numtheory): for n from 1 to 100 do if n mod 2 = 0 then m := n/ifactors(n)[2][1][1]^ifactors(n)[2][1][2] else m := n fi: printf(`%d,`,24*sigma(m)) od: # James Sellers, Dec 07 2000

a[ n_] := If[ n < 0, 0, With[ {m = Floor @ Sqrt[4 n]}, SeriesCoefficient[ Sum[ q^( x^2 + y^2 + z^2 + t^2 + (x + y + z) t ), {x, -m, m}, {y, -m, m}, {z, -m, m}, {t, -m, m}] + O[q]^(n + 1), n]]]; (* Michael Somos, Jan 11 2011 *)
a[n_] := 24*Total[ Select[ Divisors[n], OddQ]]; a[0]=1; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Sep 12 2012 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 + m) (EllipticK[ m] / (Pi/2))^2, {q, 0, n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := With[{m = InverseEllipticNomeQ @q}, SeriesCoefficient[ (1 - m/2) (EllipticK[ m] / (Pi/2))^2, {q, 0, 2 n}]]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 04 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 + EllipticTheta[ 4, 0, q]^4) / 2, {q, 0, 2 n}]; (* Michael Somos, Jun 04 2013 *)

{a(n) = if( n<1, n==0, 24 * sumdiv( n, d, d%2 * d))}; /* Michael Somos, Apr 17 2000 */

{a(n) = my(G); if( n<0, 0, G = [2, 1, 1, 1; 1, 2, 0, 0; 1, 0, 2, 0; 1, 0, 0, 2]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n, 1)), n))}; /* Michael Somos, Sep 11 2007 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^20 / (eta(x + A) * eta(x^4 + A))^8 + 16 * x * eta(x^4 + A)^8 / eta(x^2 + A)^4, n))}; /* Michael Somos, Oct 21 2017 */

from sympy import divisors
def a(n): return 1 if n==0 else 24*sum(d for d in divisors(n) if d%2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 24 2017

from math import prod
from sympy import factorint
def A004011(n): return 24*prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if p > 2) if n else 1 # Chai Wah Wu, Nov 13 2022

ModularForms( Gamma0(2), 2, prec=54).0; # Michael Somos, Jun 04 2013

a(0) = 1; if n>0 then a(n) = 24 (Sum_{d|n, d odd, d>0} d) = 24 * A000593(n).
G.f.: 1 + 24 Sum_{n>0} n x^n /(1 + x^n). a(n) = A000118(2*n) = A096727(2*n).
G.f.: (1/2) * (theta_3(z)^4 + theta_4(z)^4) = theta_3(2z)^4 + theta_2(2z)^4 = Sum_{k>=0} a(k) * x^(2*k).
G.f.: Sum_{a, b, c, d in Z} x^(a^2 + b^2 + c^2 + d^2 + a*d + b*d + c*d). - Michael Somos, Jan 11 2011
Expansion of (1 + k^2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q. - Michael Somos, Jun 10 2006
Expansion of (1 - k^2/2) * K(k^2)^2 / (Pi/2)^2 in powers of nome q^2. - Michael Somos, Mar 14 2012
Expansion of b(x) * b(x^2) + 3 * c(x) * c(x^2) in powers of x where b(), c() are cubic AGM theta functions. - Michael Somos, Jan 11 2011
Expansion of b(x) * b(x^2) + c(x) * c(x^2) / 3 in powers of x^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Mar 14 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (2 t)) = 2 (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 11 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 2*u*v - 7*v^2 - 8*v*w + 16*w^2. - Michael Somos, May 29 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2 + 4*u2^2 + 9*u3^2 + 36*u6^2 - 2*u1*u2 - 10*u1*u3 + 10*u1*u6 + 10*u2*u3 - 40*u2*u6 - 18*u3*u6. - Michael Somos, Sep 11 2007
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2 = 9.869604... (A002388). - Amiram Eldar, Dec 29 2023

Additional comments from Barry Brent (barryb(AT)primenet.com)

cp's OEIS Frontend

A004016 Theta series of planar hexagonal lattice A_2.

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A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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A004009 Expansion of Eisenstein series E_4(q) (alternate convention E_2(q)); theta series of E_8 lattice.

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Formula

A030203 Expansion of q^(-1/6) * eta(q) * eta(q^3) in powers of q.

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Magma

Mathematica

PARI

PARI

Formula

A033687 Theta series of hexagonal lattice A_2 with respect to deep hole divided by 3.

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