A112604 -id:A112604 - 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

A112605 Number of representations of n as a sum of a square and six times a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
G.f. = 1 + 2*x + 2*x^4 + x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^15 + 2*x^16 + ... - _Michael Somos_, Aug 11 2009
G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. 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
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - Michael Somos, Aug 11 2009

a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* Michael Somos, May 20 2006 */

{a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* Michael Somos, May 20 2006 */

a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos, May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos, May 20 2006
a(n)=A002324(4n+3). - Michael Somos, May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, May 20 2006, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164273. - Michael Somos, Aug 11 2009
a(3*n + 2) = 0. - Michael Somos, Aug 11 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

A112609 Number of representations of n as a sum of three times a triangular number and four times a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

			a(30) = 2 since we can write 30 = 3*10 + 4*0 = 3*6 + 4*3
q^7 + q^31 + q^39 + q^63 + q^79 + q^103 + q^111 + q^127 + q^151 + ...

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

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

A131962(n) = a(3*n). A112607(n) = a(3*n+1). A128617(n) = a(4*n+3).

A112605(2*n+1) = 2 * a(n). A112607(3*n+1) = a(n). A033762(4*n+3) = 2 * a(n). A112604(6*n+5) = 2 * a(n). A002324(8*n+7) = a(n). A123484(24*n+21) = 2 * a(n).

A112609[n_] := SeriesCoefficient[(QPochhammer[q^6]*QPochhammer[q^8])^2/
(QPochhammer[q^3]*QPochhammer[q^4]), {q,0,n}]; Table[A112609[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)

{a(n) = if( n<0, 0, n=8*n+7; sumdiv(n, d, kronecker(-3, d))/2)} /* Michael Somos, Mar 10 2008 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^8 + A))^2 / (eta(x^3 + A) * eta(x^4 + A)), n))} /* Michael Somos, Mar 10 2008 */

a(n) = 1/2*( d_{1, 3}(8n+7) - d_{2, 3}(8n+7) ) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of phi(q^3) * psi(q^4) in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 10 2008
Expansion of q^(-7/8) * (eta(q^6) * eta(q^8))^2 / (eta(q^3) * eta(q^4)) in powers of q. - Michael Somos, Mar 10 2008
Euler transform of period 24 sequence [ 0, 0, 1, 1, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 0, -2, ...]. - Michael Somos, Mar 10 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138270.
a(3*n+2) = 0.

A112606 Number of representations of n as a sum of six times a square and a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

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

The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020

			1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

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

Cf. A112604, A112608, A123484, A131961.

a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)

{a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, d)))} /* Michael Somos, Sep 29 2006 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

a(n) = d_{1, 3}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006
a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).

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

A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 28 2006, Apr 04 2008

Expansion of (a(q) - 2 * a(q^2) - a(q^4) + 2*a(q^8)) / 6 in powers of q where a() is a cubic AGM function.

			q - 2*q^2 + q^3 - 2*q^6 + 2*q^7 + q^9 + 2*q^13 - 4*q^14 - 2*q^18 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

A033762(n) = a(2*n+1). A112604(n) = a(4*n+1). -2 * A033762(n) = a(4*n+2). A112605(n) = a(4*n+3). A097195(n) = a(6*n+1). A112606(n) = a(8*n+1). -2 * A112604(n) = a(8*n+2). A112608(n) = a(8*n+3). 2 * A112607(n) = a(8*n+5). -2 * A112605(n) = a(8*n+6). 2 * A112609(n) = a(8*n+7).
A123884(n) = a(12*n+1). 2 * A121361(n) = a(12*n+7). A131961(n) = a(24*n+1). 2 * A131962(n) = a(24*n+7). A112608(n) = a(24*n+9). 2 * A131963(n) = a(24*n+13). 2 * A131964(n) = a(24*n+19).
Cf. A093766, A136748.

QP = QPochhammer; s = QP[x]^2*QP[x^6]^4*QP[x^8]*(QP[x^24]/(QP[x^2]*QP[x^3]* QP[x^12])^2) + O[x]^105; CoefficientList[s, x] (* Jean-François Alcover, Nov 06 2015, adapted from PARI, updated Dec 06 2015 *)

{a(n)=if(n<1, 0, sumdiv(n, d, if(d%2, 1, d/2%2*-2)*kronecker(-12, n/d)))}

{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==2, -2*(e<2), if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))}

Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -2, -2, -1, 0, 0, -2, 0, -2, 0, 0, -1, -2, -2, -2, 0, 0, 0, -2, -2, ...].
Moebius transform is period 24 sequence [ 1, -3, 0, 2, -1, 0, 1, 0, 0, 3, -1, 0, 1, -3, 0, 0, -1, 0, 1, -2, 0, 3, -1, 0, ...].
a(n) is multiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(3^e) = 1, a(p^e) = e+1 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 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A136748.
G.f.: x * Product_{k>0} (1 -x^(6*k)) * (1 - x^k + x^(2*k))^2 * (1 - x^(8*k)) * (1 + x^(12*k)) / (1 + x^(6*k)).
a(4*n) = a(6*n + 4) = a(6*n + 5) = 0. a(3*n) = a(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 22 2024

A107760 Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 24 2005

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 + 3*q + 3*q^2 + 3*q^3 + 3*q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 80, Eq. (32.42).

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

Cf. A033687, A033762, A035178, A097195, A112604, A112605, A123330, A132973, A132979.
Cf. A112606, A112607, A112608, A112609. - Michael Somos, Sep 27 2013
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

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

a[ n_] := If[ n < 1, Boole[n == 0], 3 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 / (4 EllipticTheta[ 2, 0, q^(3/2)]), {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
QP = QPochhammer; s = QP[q^3]*(QP[q^2]^6/(QP[q]^3*QP[q^6]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)

{a(n) = if( n<1, n==0, 3 * direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X)))[n])};

{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), n))};

{a(n) = if ( n<1, n==0, 3 * sumdiv( n, d, kronecker( -12, d)))};

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 3*A[1] # Michael Somos, Sep 27 2013

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 + u^2*w + 4 * v*w^2 - 4 * v^2*w - 2 * u*v*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 - u2) * (u1 - u2 - u3 + u6) - 3 * u6 * (u2 - u6).
Expansion of psi(q)^3 / psi(q^3) in powers of q where psi() is a Ramanujan theta function.
Expansion of (a(q) + a(q^2)) / 2 = b(q^2)^2 / b(q) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Aug 30 2008
Euler transform of period 6 sequence [ 3, -3, 2, -3, 3, -2, ...].
Moebius transform is period 6 sequence [ 3, 0, 0, 0, -3, 0, ...]. - Michael Somos, Aug 11 2009
a(n) = 3 * b(n) unless n=0 and b() is multiplicative with b(p^e) = 1 if p=2 or p=3; b(p^e) = 1+e if p == 1 (mod 6); b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6). - Michael Somos, Aug 11 2009
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (27/4)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123330. - Michael Somos, Aug 11 2009
G.f.: (Product_{k>0} (1 - x^(2*k)) / (1 - x^(2*k - 1)))^3 / (Product_{k>0} (1 - x^(6*k)) / (1 - x^(6*k - 3))). - Michael Somos, Aug 11 2009
a(n) = 3 * A035178(n) unless n=0. a(n) = (-1)^n * A132973. a(2*n) = a(3*n) = a(n). a(6*n + 5) = 0. a(2*n + 1) = 3 * A033762. a(3*n + 1) = 3 * A033687(n). a(4*n + 1) = 3 * A112604(n). a(4*n + 3) = 3 * A112605(n). a(6*n + 1) = 3 * A097195(n). Convolution inverse of A132979.
a(8*n + 1) = 3 * A112606(n). a(8*n + 3) = 3* A112608(n). a(8*n + 5) = 6 * A112607(n-1). a(8*n + 7) = 6 * A112609(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi*sqrt(3)/2 = 2.720699... . - Amiram Eldar, Dec 28 2023

A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 12, -6, -6, 0, 12, -12, 12, -6, 0, 0, -6, -12, 24, 0, -6, 0, 12, -12, 0, -12, 0, 0, 12, -6, 24, -6, -12, 0, 0, -12, 12, 0, 0, 0, -6, -12, 24, -12, 0, 0, 24, -12, 0, 0, 0, 0, -6, -18, 12, 0, -12, 0, 12, 0, 24, -12, 0, 0, 0, -12, 24, -12, -6, 0, 0, -12, 0, 0, 0, 0, 12, -12, 24, -6, -12, 0, 24, -12, 0, -6, 0, 0, -12, 0, 24

Offset: 0

Michael Somos, Sep 15 2006

sign

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

			G.f. = 1 - 6*q + 12*q^2 - 6*q^3 - 6*q^4 + 12*q^6 - 12*q^7 + 12*q^8 - 6*q^9 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

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

Cf. A033687, A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A113660, A121361, A122860, A123884, A227354.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 / EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Sep 27 2013 *)

{a(n)= if( n<1, n==0, 6 * sumdiv(n, d, (-1)^(n/d) * kronecker( -3, d)))}

{a(n)= if( n<1, n==0, -6 * sumdiv(n, d, (2 + (-1)^d) * kronecker( -3, d)))}

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

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] - 6 *A[1] # Michael Somos, Sep 27 2013

Expansion of 2*a(q^2) - a(q) = b(q)^2 / b(q^2) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [ -6, -3, -4, -3, -6, -2, ...].
Moebius transform is period 6 sequence [ -6, 18, 0, -18, 6, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v*(u+v)^2 - 2*u*w*(v+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-u2-u3+u6) * (u1+2*u2+u3) - (2*u1+u2-2*u3-u6) * (u1+2*u2-u3).
G.f.: Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3k)) = 1 + 6 * Sum_{k>0} (-1)^k * x^k / (1 + x^k + x^(2*k)).
G.f.: 1 - 6 * (Sum_{k>0} x^(3*k - 2) / (1 + x^(3*k - 2)) - x(3*k - 1)
/ (1 + x^(3*k - 1))).
a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.
(-1)^n * a(n) = A113660(n). -6 * a(n) = A122860(n) if n>0.
a(2*n) = A227354(n). a(2*n + 1) = -6 * A033762(n). a(3*n + 1) = -6 * A033687(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 3) = -6 * A112605(n). a(6*n + 1) = -6 * A097195(n). a(8*n + 1) = -6 * A112606(n). a(8*n + 3) = -6 * A112608(n). a(8*n + 5) = -12 * A112607(n-1). a(8*n + 7) = -12 * A112609(n). a(12*n + 1) = -6 * A123884(n). a(12*n + 7) = -12 * A121361(n). - Michael Somos, Sep 27 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - Amiram Eldar, Nov 23 2023

A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

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, Nov 02 2005

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
Li-Chien Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1105, equation (3.8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A073010, A093829, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964. A133637.

Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

A := Basis( ModularForms( Gamma1(24), 1), 106); A[2] + A[3] + A[4] - A[5] + A[7] + 2*A[8] + A[9] + A[10]; /* Michael Somos, May 07 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, Jan 31 2015 *)

{a(n) = if( n<1, 0, -(-1)^max( 1, valuation( n, 2)) * sumdiv(n, d, kronecker( -12, d)))};

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

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^3 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^3), n))};

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 0, -2, -1, 0, 1, 2, 0, 0,-1][d%12 + 1]))}; /* Michael Somos, May 07 2015 */

Expansion of (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-1)^e if e>0, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - 2 * x^(12*k - 8) / (1 - x^(12*k - 8)) + 2 * x^(12*k - 4) / (1 - x^(12*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(3*k))^2 / (1 + x^(4*k) + x^(8*k)).
G.f.: x * Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)) * ((1 - x^(12*k - 6)) / (1 - x^(3*k)))^3.
Expansion of theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132973.
a(n) = -(-1)^n * A093829(n). - Michael Somos, Jan 31 2015
Convolution inverse of A133637.
a(3*n) = a(n). a(6*n + 5) = a(12*n + 10) = 0. |a(n)| = A035178(n).
a(2*n) = A093829(n). a(2*n + 1) = A033762(n).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 6) = A112605(n). a(8*n + 7) = 2 * A112609(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 * A121963(n). a(24*n + 19) = 2 * A131964(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 23 2023

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A004016 Theta series of planar hexagonal lattice A_2.

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A112605 Number of representations of n as a sum of a square and six times a triangular number.

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

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A112609 Number of representations of n as a sum of three times a triangular number and four times a triangular number.

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A112606 Number of representations of n as a sum of six times a square and a triangular number.

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