A033687 -id:A033687 - 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

A005882 Theta series of planar hexagonal lattice (A2) with respect to deep hole.

Original entry on oeis.org

3, 3, 6, 0, 6, 3, 6, 0, 3, 6, 6, 0, 6, 0, 6, 0, 9, 6, 0, 0, 6, 3, 6, 0, 6, 6, 6, 0, 0, 0, 12, 0, 6, 3, 6, 0, 6, 6, 0, 0, 3, 6, 6, 0, 12, 0, 6, 0, 0, 6, 6, 0, 6, 0, 6, 0, 9, 6, 6, 0, 6, 0, 0, 0, 6, 9, 6, 0, 0, 6, 6, 0, 12, 0, 6, 0, 6, 0, 0, 0, 6, 6, 12, 0, 0, 3, 12, 0, 0, 6, 6, 0, 6, 0, 6, 0, 3, 6, 0, 0, 12

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.
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).
On page 111 of Conway and Sloane is "If the origin is moved to a deep hole the theta series is Theta_{hex+[1]}(z) = theta_2(z) psi_6(3z) + theta_3(z) psi_3(3z) = 3 q^{1/3} + 3 q^{4/3} + 6 q^{7/3} + 6 q^{13/3} + ... (63)" where the psi_k() for integer k is defined on page 103 equation (11) as psi_k(z) = e^{Pi i/z^2} theta_3(Pi z/k | z) = Sum_{m in Z} q^{(m + 1/k)^2}. - Michael Somos, Sep 10 2018

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

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111.
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.
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.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Essentially same as A033685 and A033687.

Cf. A113062, A273845.

Basis( ModularForms( Gamma1(9), 1), 302)[2] * 3; /* Michael Somos, Jul 19 2014 */

a[ n_] := SeriesCoefficient[ 3 QPochhammer[ q^3]^3 / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Jul 19 2014 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 3 * eta(x^3 + A)^3 / eta(x + A), n))}; /* Michael Somos, Aug 15 2006 */

Expansion of q^(-1/3) * 3 * eta(q^3)^3 / eta(q) in powers of q.
Expansion of q^(-1/3) * c(q) in powers of q where c(q) is the third cubic AGM theta function.
Given g.f. A(x), then B(x) = x*A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 2*u*w^2 - u^2*w. - Michael Somos, Aug 15 2006
G.f.: 3 Product_{k>0} (1-q^(3k))^3/(1-q^k).
G.f.: Sum_{u,v in Z} x^(u*u + u*v + v*v + u + v). - Michael Somos, Jul 19 2014
a(n) = 3 * A033687(n). a(n) = A113062(3*n + 1) = A033685(3*n + 1).
Expansion of 2 * psi(x^2) * f(x^2, x^4) + phi(x) * f(x^1, x^5) in powers of x 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/sqrt(3). - Vaclav Kotesovec, Dec 17 2022

A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -3. See Formula section for the general expression. - N. J. A. Sloane, Mar 22 2022
Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p) * p^(-2s))^(-1) for m = -3.
(Number of points of norm n in hexagonal lattice) / 6, n>0.
The hexagonal lattice is the familiar 2-dimensional lattice (A_2) in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The first occurrence of a(n) = 1, 2, 3, 4,... is at n= 1, 7, 49, 91, 2401, 637, ... as tabulated in A343771. - R. J. Mathar, Sep 21 2024

			G.f. = x + x^3 + x^4 + 2*x^7 + x^9 + x^12 + 2*x^13 + x^16 + 2*x^19 + 2*x^21 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.
J. W. L. Glaisher, Table of the excess of the number of (3k+1)-divisors of a number over the number of (3k+2)-divisors, Messenger Math., 31 (1901), 64-72.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 1..10000
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Hershel M. Farkas, On an arithmetical function, Ramanujan J., 8(3) (2004), 309-315.
Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, arXiv:1905.06506 [math.NT], 2019.
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.
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. [From _N. J. A. Sloane_, Mar 12 2009]
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009
N. J. A. Sloane, Maple code for Dedekind zeta functions of quadratic number fields.

Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.
Cf. A004016, A035019, A073010, A145377, A293899, A000086, A003136, A034020, A145394.

a002324 n = a001817 n - a001822 n  -- Reinhard Zumkeller, Nov 26 2011

A002324 := proc(n)
    local a,pe,p,e;
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 3 then
            ;
        elif modp(p,3) = 1 then
            a := a*(e+1) ;
        else
            a := a*(1+(-1)^e)/2 ;
        end if;
    end do:
    a ;
end proc:
seq(A002324(n),n=1..100) ; # R. J. Mathar, Sep 21 2024

dn12[n_]:=Module[{dn=Divisors[n]},Count[dn,?(Mod[#,3]==1&)]-Count[ dn,?(Mod[#,3]==2&)]]; dn12/@Range[120]  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Aug 24 2014 *)
Table[DirichletConvolve[DirichletCharacter[3,2,m],1,m,n],{n,1,30}] (* Steven Foster Clark, May 29 2019 *)
f[3, p_] := 1; f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1+(-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

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

{a(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)))};

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

{a(n) = if( n<1, 0, qfrep([2,1; 1,2], n, 1)[n] / 3)}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-3, p)*X))[n])}; \\ Michael Somos, Jun 05 2005

my(B=bnfinit(x^2+x+1)); vector(100,n,#bnfisintnorm(B,n)) \\ Joerg Arndt, Jun 01 2024

from math import prod
from sympy import factorint
def A002324(n): return prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) # Chai Wah Wu, Nov 17 2022

From N. J. A. Sloane, Mar 22 2022 (Start):
The Dedekind zeta function DZ_K(s) for a quadratic field K of discriminant D is as follows.
Here m is defined by K = Q(sqrt(m)) (so m=D/4 if D is a multiple of 4, otherwise m=D).
DZ_K(s) is the product of three terms:
(a) Product_{odd primes p | D} 1/(1-1/p^s)
(b) Product_{odd primes p such that (D|p) = -1} 1/(1-1/p^(2s))
(c) Product_{odd primes p such that (D|p) = 1} 1/(1-1/p^s)^2
and if m is
   0,1,2,3,4,5,6,7 mod 8, the prime 2 is to be included in term
   -,c,a,a,-,b,a,a, respectively.
For Maple (and PARI) implementations, see link. (End)
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 + 4*w^2 - 2*u*w + w - v. - Michael Somos, Jul 20 2004
Has a nice Dirichlet series expansion, see PARI line.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic, Dec 16 2002
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = A033687(n). - Michael Somos, Apr 04 2003
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
Multiplicative with a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3). - Michael Somos, May 20 2005
G.f.: Sum_{k>0} x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1)  / (1 - x^(3*k - 1)). - Michael Somos, Nov 02 2005
G.f.: Sum_{n >= 1} q^(n^2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009
a(n) = A001817(n) - A001822(n). - R. J. Mathar, Mar 31 2011
A004016(n) = 6*a(n) unless n=0.
Dirichlet g.f.: zeta(s)*L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3 (A102283). - Ralf Stephan, Mar 27 2015
From Andrey Zabolotskiy, May 07 2018: (Start)
a(n) = Sum_{ m: m^2|n } A000086(n/m^2).
a(A003136(m)) > 0, a(A034020(m)) = 0 for all m. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 11 2022

More terms from David Radcliffe

Somos D.g.f. replaced with correct version by Ralf Stephan, Mar 27 2015

A097195 Expansion of s(12)^3s(18)^2/(s(6)^2s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 16 2004

			G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
G.f. = q + 2*q^7 + 2*q^13 + 2*q^19 + q^25 + 2*q^31 + 2*q^37 + 2*q^43 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.38).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A004016, A010815, A093602.

Cf. A008441, A033687, A340456, A374900.

a[n_] := DivisorSum[6n+1, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 100} ] (* Jean-François Alcover, Nov 23 2015, after Michael Somos *)
QP = QPochhammer; s = QP[q^2]^3*(QP[q^3]^2/QP[q]^2/QP[q^6]) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(6 n + 1))]; (* Michael Somos, Mar 05 2016 *)

{a(n) = if( n<0, 0, sumdiv(6*n+1, d, kronecker(-3, d)))}; /* Michael Somos, Nov 03 2005 */

{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>3, if( p%6==1, e+1, !(e%2)))))}; /* Michael Somos, Nov 03 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))}; /* Michael Somos, Nov 03 2005 */

Fine gives an explicit formula for a(n) in terms of the divisors of n.
a(n) = b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
From Michael Somos, Nov 03 2005: (Start)
G.f.: Sum_{k in Z} x^k / (1 - x^(6*k + 1)).
G.f.: Sum_{k>=0} a(k) * x^(6*k + 1) = Sum_{k>0} x^(2*k-1) * (1 - x^(4*k - 2)) * (1 - x^(8*k - 4)) * (1 - x^(20*k - 10)) / (1 - x^(36*k - 18)). (End)
From Michael Somos, Mar 05 2016: (Start)
Expansion of q^(-1/6) * eta(q^2)^3 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 2, -1, 0, -1, 2, -2, ...].
6 * a(n) = A004016(6*n + 1). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Nov 24 2023

A000731 Expansion of Product (1 - x^k)^8 in powers of x.

Original entry on oeis.org

1, -8, 20, 0, -70, 64, 56, 0, -125, -160, 308, 0, 110, 0, -520, 0, 57, 560, 0, 0, 182, -512, -880, 0, 1190, -448, 884, 0, 0, 0, -1400, 0, -1330, 1000, 1820, 0, -646, 1280, 0, 0, -1331, -2464, 380, 0, 1120, 0, 2576, 0, 0, -880, 1748, 0, -3850, 0, -3400, 0, 2703, 4160, -2500, 0, 3458

Offset: 0

N. J. A. Sloane

Number 22 of the 74 eta-quotients listed in Table I of Martin (1996).
Denoted by g_4(q) in Cynk and Hulek in Remark 3.4 on page 12 as the unique level 9 form of weight 4.
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
a(n)=0 if and only if A033687(n)=0 (see the Han-Ono paper). - Emeric Deutsch, May 16 2008
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 8*x + 20*x^2 - 70*x^3 + 64*x^4 + 56*x^5 - 125*x^6 - 160*x^7 + ...
G.f. = q - 8*q^4 + 20*q^7 - 70*q^13 + 64*q^16 + 56*q^19 - 125*q^25 - ...

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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
Matthew Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Shaun Cooper, Minchael D. Hirschhorn and Richard Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
Slawomir Cynk and Klaus Hulek, Construction and examples of higher-dimensional modular Calabi-Yau manifolds, arXiv:math/0509424 [math.AG], 2005-2006.
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
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.
Iva Kodrnja and Helena Koncul, Polynomials vanishing on a basis of S_m(Gamma_0(N)), Glasnik Matematički (2024) Vol. 59, No. 79, 313-325. See p. 319.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Morris Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A033687, A033690, A092342, A153728.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A002107, A010815 - A010840.

Basis( CuspForms( Gamma0(9), 4), 56) [1]; /* Michael Somos, Dec 09 2013 */

a[ n_] := SeriesCoefficient[ QPochhammer[ x]^8, {x, 0, n}]; (* Michael Somos, Sep 29 2011 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^8, {x, 0, n}]; (* Michael Somos, Dec 09 2013 *)

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^8, n))};

{a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p%3==2, if( e%2, 0, (-1)^(e/2) * p^(3*e/2)), forstep( y=sqrtint(4*p\3), sqrtint(p\3), -1, if( issquare( 4*p - 3*y^2, &x), if( x%3!=2, x=-x); break)); a0=1; a1 = y = x * (x^2 - 3*p); for( i=2, e, x = y*a1 - p^3*a0; a0=a1; a1=x); a1)))}; /* Michael Somos, Aug 23 2006 */

CuspForms( Gamma0(9), 4, prec=56).0; # Michael Somos, May 28 2013

Expansion of q^(-1/3) * eta(q)^8 in powers of q.
Expansion of q^(-1/3) * b(q)^3 * c(q) / 3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Expansion of q^(-1) * b(q) * c(q)^3 / 27 in powers of q^3 where b(), c() are cubic AGM theta functions. - Michael Somos, Nov 08 2006
Euler transform of period 1 sequence [ -8, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-1)^(e/2) * p^(3*e/2) if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) - b(p^(e-2))*p^3 if p == 1 (mod 3) where b(p) = (x^2 - 3*p)*x, 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Michael Somos, Aug 23 2006
Given g.f. A(x), then B(x) = x * A(x^3) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 - u * w * (u + 16 * w). - Michael Somos, Feb 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = 81 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 29 2011
G.f.: Product_{k>0} (1 - x^k)^8.
a(2*n) = A153728(n). - Michael Somos, Sep 29 2011
a(4*n + 1) = -8 * a(n). - Michael Somos, Dec 06 2004
a(4*n + 3) = a(16*n + 13) = 0. - Michael Somos, Oct 19 2005
A092342(n) = a(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{n>=0} a(n) * q^(3*n + 1) = (Sum_{i,j,k in Z} (i-j) * (j-k) * (k-i) * q^((i*i + j*j + k*k) / 2)) / 2 where 0 = i+j+k, i == 1 (mod 3), j == 2 (mod 3), and k == 0 (mod 3). - Michael Somos, Sep 22 2014
a(0) = 1, a(n) = -(8/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 27 2017
G.f.: exp(-8*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
Let M = p_1*...*p_k be a positive integer whose prime factors p_i (not necessarily distinct) are all congruent to 2 (mod 3). Then a( M^2*n + (M^2 - 1)/3 ) = (-1)^k*M^3*a(n). See Cooper et al., Theorem 1. - Peter Bala, Dec 01 2020
a(n) = b(3*n + 1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p^3)^(e/2) if p == 2 (mod 3), b(p^e) = (((x+sqrt(-3)*y)/2)^(3*e+3) - ((x-sqrt(-3)*y)/2)^(3*e+3))/(((x+sqrt(-3)*y)/2)^3 - ((x-sqrt(-3)*y)/2)^3) if p == 1 (mod 3) where 4*p = x^2 + 3*y^2, |x|<|y| and x == 2 (mod 3). - Jianing Song, Mar 19 2022

Corrected by Charles R Greathouse IV, Sep 02 2009

A093829 Expansion of q * psi(q^3)^3 / psi(q) in powers of q where psi() is a Ramanujan theta function.

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

Michael Somos, Apr 17 2004

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. = q - q^2 + q^3 + q^4 - q^6 + 2*q^7 - q^8 + q^9 + q^12 + 2*q^13 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025. See p. 3.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A112300, A113447, A122859.
Cf. A097195, A112604, A112605, A112606, A112607, A112608. - Michael Somos, Sep 27 2013
Cf. A000700, A000122, A010054, A121373.

Basis( ModularForms( Gamma1(6), 1), 90) [2]; /* Michael Somos, Jul 02 2014 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 2, -1, 0} [[ Mod[#, 6, 1]]] &]];
QP = QPochhammer; s = (QP[q]*QP[q^6]^6)/(QP[q^2]^2*QP[q^3]^3) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

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

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

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d) - if( d%2==0, 2 * kronecker( -3, d/2) ) ))}; /* Michael Somos, May 29 2005 */

ModularForms( Gamma1(6), 1, prec=90).1; # Michael Somos, Sep 27 2013

Expansion of (a(q) - a(q^2)) / 6 = c(q^2)^2 / (3 * c(q)) in powers of q where a(), c() are cubic AGM functions. - Michael Somos, Sep 06 2007
Expansion of (eta(q) * eta(q^6)^6) / (eta(q^2)^2 * eta(q^3)^3) in powers of q.
Euler transform of period 6 sequence [ -1, 1, 2, 1, -1, -2, ...].
Moebius transform is period 6 sequence [ 1, -2, 0, 2, -1, 0, ...] = A112300. - Michael Somos, Jul 16 2006
Multiplicative with a(p^e) = (-1)^e if p=2; a(p^e) = 1 if 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. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 12^(-1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A122859.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 4*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) = u2 * (u2 - u3 - 4*u6) - (u3 + u6) * (u1 - 3*u3 - 3*u6).
G.f.: Sum_{k>0} (x^k - 2 * x^(2*k) + 2 * x^(4*k) - x^(5*k)) / (1 - x^(6*k)) = x * Product_{k>0} ((1 - x^k) * (1 - x^(6*k))^6) / ((1 - x^(2*k))^2 * (1 - x^(3*k))^3).
a(n) = -(-1)^n * A113447(n). - Michael Somos, Jan 31 2015
a(2*n) = -a(n). a(3*n) = a(n). a(6*n + 5) = 0.
A035178(n) = |a(n)|. A033762(n) = a(2*n + 1). A033687(n) = a(3*n + 1).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.302299894039... . - Amiram Eldar, Nov 21 2023

A033686 One-ninth of theta series of A2[hole]^2.

Original entry on oeis.org

1, 2, 5, 4, 8, 6, 14, 8, 14, 10, 21, 16, 20, 14, 28, 16, 31, 18, 40, 20, 32, 28, 42, 24, 38, 32, 62, 28, 44, 30, 56, 40, 57, 34, 70, 36, 72, 38, 70, 48, 62, 52, 85, 44, 68, 46, 112, 56, 74, 50, 100, 64, 80, 64, 98, 56, 108, 58, 124, 60, 112, 76, 112, 64, 98, 66, 155, 80, 104

Offset: 0

N. J. A. Sloane

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

Number of partition pairs of n where each partition is 3-core (see Theorem 2.1 of Wang link). - Michel Marcus, Jul 14 2015

			G.f. = 1 + 2*x + 5*x^2 + 4*x^3 + 8*x^4 + 6*x^5 + 14*x^6 + 8*x^7 + 14*x^8 + ...
G.f. = q^2 + 2*q^5 + 5*q^8 + 4*q^11 + 8*q^14 + 6*q^17 + 14*q^20 + 8*q^23 + ...
Theta series of A2[hole]^2 = c(q)^2 = 9*q^(2/3) + 18*q^(5/3) + 45*q^(8/3) + 36*q^(11/3) + 72*q^(14/3) + 54*q^(17/3) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 111, Eq (63)^2.

Muniru A Asiru, Table of n, a(n) for n = 0..20000
Kevin B. Ford, Some infinite series identities, Proc. Amer. Math. Soc., Volume 119, Number 3 (November 1993), 1019-1020.
Liuquan Wang, Explicit Formulas for Partition Pairs and Triples with 3-Cores, arXiv:1507.03099 [math.NT], 2015.

Cf. A033687, A096726, A097723, A134079, A144615, A242874, A281722.

Cf. A000203 (sigma), A016789 (3n+2).

sequence := List([1..100010],n->Sigma(3*n-1)/3); # Muniru A Asiru, Dec 29 2017

Basis( ModularForms( Gamma0(9), 2), 195)[3]; /* Michael Somos, Jul 14 2015 */

[SumOfDivisors(3*n+2)/3: n in [0..70]]; // Vincenzo Librandi, Jan 13 2018

with(numtheory): seq(sigma(3*n-1)/3, n=1..2000); # Muniru A Asiru, Jan 18 2018

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3]^3 / QPochhammer[ x])^2, {x, 0, n}]; (* Michael Somos, May 26 2014 *)
Array[ DivisorSigma[1, 3 # - 1]/3 &, 69] (* Robert G. Wilson v, Jan 12 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A)^3 / eta(x + A))^2, n))}; /* Michael Somos, Oct 17 2006 */

a(n)=sigma(3*n+2)/3; \\ Michel Marcus, Jul 14 2015

ModularForms( Gamma0(9), 2, prec=195).2 # Michael Somos, May 26 2014

a(n) = sigma(3*n+2)/3. Euler transform of period 3 sequence [2, 2, -4, ...]. - Vladeta Jovovic, Sep 14 2004
Expansion of q^(-2/3) * c(q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function. - Michael Somos, Oct 17 2006
Expansion of q^(-2/3) * (eta(q^3)^3 / eta(q))^2 in powers of q. - Michael Somos, Mar 16 2012
Convolution square of A033687. - Michael Somos, Oct 17 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (9 t)) = (1/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A242874.
27 * a(n) = A096726(3*n + 2) - A281722(3*n + 2). - Michael Somos, Sep 04 2017
a(n) = A144615(n)/3. - Robert G. Wilson v, Jan 12 2018
From Peter Bala, Jan 07 2021: (Start)
a(n) = (-1)^n*A134079(n).
A(x) = Sum_{n = -oo..oo} x^(2*n)/(1 - x^(3*n+1))^2 = Sum_{n = -oo..oo} x^(4*n+2)/(1 - x^(3*n+2))^2 (apply Ford, equation 1, with c = x^(3/2), d = x^(1/2), |x| < 1 to the g.f. Sum_{n = -oo..oo} x^n /(1 - x^(3*n + 1)) of A033687).
Conjectural g.f.: A(x) = Sum_{n = -oo..oo} x^n/(1 - x^(3*n+2))^2 = Sum_{n = -oo..oo} x^(5*n+1)/(1 - x^(3*n+1))^2. (End)
Sum_{k=1..n} a(k) = (2*Pi^2/27) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

A175595 Square array A(n,t), n>=0, t>=0, read by antidiagonals: A(n,t) is the number of t-core partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 0, 5, 1, 1, 2, 1, 0, 7, 1, 1, 2, 0, 0, 0, 11, 1, 1, 2, 3, 2, 0, 0, 15, 1, 1, 2, 3, 1, 1, 1, 0, 22, 1, 1, 2, 3, 5, 3, 2, 0, 0, 30, 1, 1, 2, 3, 5, 2, 3, 0, 0, 0, 42, 1, 1, 2, 3, 5, 7, 6, 3, 1, 0, 0, 56, 1, 1, 2, 3, 5, 7, 5, 5, 4, 2, 1, 0, 77, 1, 1, 2, 3, 5, 7, 11, 9, 7, 4, 2, 0, 0, 101

Offset: 0

Alois P. Heinz, Dec 03 2010

A partition of n is a t-core partition if none of the hook numbers associated to the Ferrers-Young diagram is a multiple of t. See Chen reference for definitions.

			A(4,3) = 2, because there are 2 partitions of 4 such that no hook number is a multiple of 3:
   (1)  2 | 4 1
       +1 | 2
       +1 | 1
   -------+-----
   (2)  3 | 4 2 1
       +1 | 1
Square array A(n,t) begins:
   1,  1,  1,  1,  1,  1,  1,  1,  ...
   1,  0,  1,  1,  1,  1,  1,  1,  ...
   2,  0,  0,  2,  2,  2,  2,  2,  ...
   3,  0,  1,  0,  3,  3,  3,  3,  ...
   5,  0,  0,  2,  1,  5,  5,  5,  ...
   7,  0,  0,  1,  3,  2,  7,  7,  ...
  11,  0,  1,  2,  3,  6,  5, 11,  ...
  15,  0,  0,  0,  3,  5,  9,  8,  ...

Garvan, F. G., A number-theoretic crank associated with open bosonic strings. In Number Theory and Cryptography (Sydney, 1989), 221-226, London Math. Soc. Lecture Note Ser., 154, Cambridge Univ. Press, Cambridge, 1990.
James, Gordon; and Kerber, Adalbert, The Representation Theory of the Symmetric Group. Addison-Wesley Publishing Co., Reading, Mass., 1981.

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, arXiv:math/0208050 [math.NT], 2002.
A. O. L. Atkins and F. G. Garvan, Relations between the ranks and cranks of partitions, Rankin memorial issues. Ramanujan J. 7 (2003), 343-366.
Shichao Chen, Arithmetical properties of the number of t-core partitions, The Ramanujan Journal, 18 (2007), no. 1, 103-112, DOI: 10.1007/s11139-007-9045-5.
F. G. Garvan, The crank of partitions mod 8, 9 and 10, Trans. Amer. Math. Soc. 322 (1990), 79-94.
F. G. Garvan, Some congruences for partitions that are p-cores, Proc. London Math. Soc. 66 (1993), 449-478.
F. G. Garvan, More cranks and t-cores, Bull. Austral. Math. Soc. 63 (2001), 379-391.
F. G. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.
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.
Ben Kane, Sums of Triangular Numbers and t-Core Partitions, Journal of Combinatorics and Number Theory, 1 (2009), no.1, 59-64.
B. Kim, On inequalities and linear relations for 7-core partitions, Discrete Math., 310 (2010), 861-868.
N. J. A. Sloane, Transforms.

Columns t=0-12 give A000041, A000007, A010054, A033687, A045831, A053723, A081622, A053724, A182803, A182804, A182805, A053691, A192061.
Rows n=0-1 give A000012, A060576.
Diagonal gives A000094(n+1) for n>0.
Upper diagonal gives A000041.
Lower diagonal (conjectured) gives A086642 for n>0.

with(numtheory):
A:= proc(n, t) option remember; `if`(n=0, 1,
      add(add(`if`(t=0 or irem(d, t)=0, d-d*t, d),
              d=divisors(j))*A(n-j, t), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);
(From N. J. A. Sloane, Jun 21 2011: to get M terms of the series for t-core partitions:)
M:=60;
f:=proc(t) global M; local q,i,t1;
t1:=1;
for i from 1 to M+1 do
t1:=series(t1*(1-q^(i*t))^t,q,M);
t1:=series(t1/(1-q^i),q,M);
od;
t1;
end;
# then for example seriestolist(f(5));

n = 13; f[t_] = (1-x^(t*k))^t/(1-x^k); f[0] = 1/(1-x^k);
s[t_] := CoefficientList[ Series[ Product[ f[t], {k, 1, n}], {x, 0, n}], x]; m = Table[ PadRight[ s[t], n+1], {t, 0, n}]; Flatten[ Table[ m[[j+1-k, k]], {j, n+1}, {k, j}]] (* Jean-François Alcover, Jul 25 2011, after g.f. *)

G.f. of column t: Product_{i>=1} (1-x^(t*i))^t/(1-x^i).

Column t is the Euler transform of period t sequence [1, .., 1, 1-t, ..].

Additional references from N. J. A. Sloane, Jun 21 2011

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

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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A005882 Theta series of planar hexagonal lattice (A2) with respect to deep hole.

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A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

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A097195 Expansion of s(12)^3s(18)^2/(s(6)^2s(36)), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815. Then replace q^6 with q.