A033716 -id:A033716 - 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

A025428 Number of partitions of n into 4 nonzero squares.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 3, 0, 1, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 3, 2, 1, 2, 1, 2, 1, 2, 2, 1, 4, 1, 2, 3, 0, 2, 4, 1, 3, 2, 1, 4, 1, 1, 3, 3, 2, 2, 4, 2, 1, 3, 2, 3, 4, 2, 3, 3, 1, 2, 5, 2, 4, 3, 2, 4, 1, 1, 6, 4, 3, 4, 2, 3, 0, 4, 4, 3, 5, 1, 5, 5, 1, 4, 5, 2

Offset: 0

David W. Wilson

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

A025428 := proc(n)
    local a,i,j,k,lsq ;
    a := 0 ;
    for i from 1 do
        if 4*i^2 > n then
            return a;
        end if;
        for j from i do
            if i^2+3*j^2 > n then
                break;
            end if;
            for k from j do
                if i^2+j^2+2*k^2 > n then
                    break;
                end if;
                lsq := n-i^2-j^2-k^2 ;
                if lsq >= k^2 and issqr(lsq) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc:
seq(A025428(n),n=1..40) ; # R. J. Mathar, Jun 15 2018
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
     `if`(i<1 or t<1, 0, b(n, i-1, t)+`if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2019

nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *)
f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *)

A025428(n)=sum(a=1,n,sum(b=1,a,sum(c=1,b,sum(d=1,c,a^2+b^2+c^2+d^2==n))))

A025428(n)=sum(a=1,sqrtint(max(n-3,0)), sum(b=1,min(sqrtint(n-a^2-2),a), sum(c=1,min(sqrtint(n-a^2-b^2-1),b),issquare(n-a^2-b^2-c^2,&d) & d <= c )))

A025428(n)=sum(a=sqrtint(max(n,4)\4),sqrtint(max(n-3,0)), sum(b=sqrtint((n-a^2)\3-1)+1,min(sqrtint(n-a^2-2),a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1),b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012
for(n=1,100,print1(A025428(n),","))

T(n)={a=matrix(n,4,i,j,0);for(d=1,sqrtint(n),forstep(i=n,d*d+1,-1,for(j=2,4,a[i,j]+=sum(k=1,j,if(k0,a[i-k*d*d,j-k],if(k==j&&i-k*d*d==0,1)))));a[d*d,1]=1);for(i=1,n,print(i" "a[i,4]))} /* Robert Gerbicz, Sep 28 2012 */

For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012
a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012
a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

A092573 Number of solutions to x^2 + 3y^2 = n in positive integers x and y.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Feb 28 2004

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
E. Akhtarkavan, M. F. M. Salleh and O. Sidek, Multiple Descriptions Video Coding Using Coinciding Lattice Vector Quantizer for H.264/AVC and Motion JPEG2000, World Applied Sciences Journal 21 (2): 157-169, 2013. - From _N. J. A. Sloane_, Feb 11 2013
Eric Weisstein's World of Mathematics, Euler's 6n+1 Theorem

Cf. A002476, A092572, A092575.

N:= 300: # to get a(0)..a(N)
V:= Vector(N):
for y from 1 to floor(sqrt(N/3-1)) do
  js:= [seq(x^2+3*y^2, x=1..floor(sqrt(N-3*y^2)))];
  V[js]:= map(`+`,V[js],1);
od:
0,op(convert(V,list)); # Robert Israel, Apr 03 2017

r[z_] := Reduce[x > 0 && y > 0 && x^2 + 3 y^2 == z, {x, y}, Integers]; Table[rz = r[z]; If[rz === False, 0, If[rz[[0]] === Or, Length[rz], 1]], {z, 0, 102}] (* Jean-François Alcover, Oct 23 2012 *)
gf = (EllipticTheta[3, 0, x]-1)*(EllipticTheta[3, 0, x^3]-1)/4 + O[x]^105;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 02 2018, after Robert Israel *)

a(n) = ( A033716(n) - A000122(n) - A000122(n/3) + A000007(n) )/4. - Max Alekseyev, Sep 29 2012

G.f.: (Theta_3(0,x)-1)*(Theta_3(0,x^3)-1)/4 where Theta_3 is a Jacobi theta function. - Robert Israel, Apr 03 2017

Definition corrected by David A. Corneth, Apr 03 2017

A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

Original entry on oeis.org

1, 1, 5, 19, 85, 376, 1715, 7890, 36693, 171820, 809380, 3830619, 18201235, 86770516, 414836210, 1988138644, 9548771157, 45948159420, 221470766204, 1069091485500, 5167705849460, 25009724705460, 121171296320475, 587662804774890, 2852708925078675, 13859743127937876

Offset: 0

César Eliud Lozada, Oct 17 2021

Suppose n objects are to be distributed into 2n baskets, half of these white and half black. White baskets may contain 0 or any number of objects, while black baskets may contain 0 or an even number of objects. a(n) is the number of distinct possible distributions.

			Some examples (semicolon separates white basket from black baskets):
For n=1: {{1 ; 0}} - Total possible ways: 1.
For n=2: {{0, 0 ; 0, 2}, {0, 0 ; 2, 0}, {0, 2 ; 0, 0}, {1, 1 ; 0, 0}, {2, 0 ; 0, 0}} - Total possible ways: 5.

Alois P. Heinz, Table of n, a(n) for n = 0..1441

Cf. A033715, A033716, A034297, A063020, A088218, A234839, A294860, A348474, A351856, A351857, A352373.

b:= proc(n, t) option remember; `if`(t=0, 1-signum(n),
      add(b(n-j, t-1)*(1+iquo(j, 2)), j=0..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 17 2021

(* giveList=True produces the list of solutions *)
(* giveList=False gives the number of solutions *)
counter[objects_, giveList_: False] :=
  Module[{n = objects, nb, eq1, eqa, eqb, eqs, var, sol, var2, list},
   nb = n;
   eq1 = {Total[Map[a[#] + 2*b[#] &, Range[nb]]] - n == 0};
   eqa = {And @@ Map[0 <= a[#] <= n &, Range[nb]]};
   eqb = {And @@ Map[0 <= b[#] <= n &, Range[nb]]};
   eqs = {And @@ Join[eq1, eqa, eqb]};
   var = Flatten[Map[{a[#], b[#]} &, Range[nb]]];
   var = Join[Map[a[#] &, Range[nb]], Map[b[#] &, Range[nb]]];
   sol = Solve[eqs, var, Integers];
   var2 = Join[Map[a[#] &, Range[nb]], Map[2*b[#] &, Range[nb]]];
   list = Sort[Map[var2 /. # &, sol]];
   list = Map[StringReplace[ToString[#], {"," -> " ;"}, n] &, list];
   list = Map[StringReplace[#, {";" -> ","}, n - 1] &, list];
   Return[
    If[giveList, Print["Total: ", Length[list]]; list, Length[sol]]];
   ];
(* second program: *)
b[n_, t_] := b[n, t] = If[t == 0, 1 - Sign[n], Sum[b[n - j, t - 1]*(1 + Quotient[j, 2]), {j, 0, n}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

Conjecture: D-finite with recurrence +7168*n*(2*n-1)*(n-1)*a(n) -64*(n-1)*(1759*n^2-5294*n+5112)*a(n-1) +12*(7561*n^3-75690*n^2+165271*n-101070)*a(n-2) +5*(110593*n^3-743946*n^2+1659971*n-1232778)*a(n-3) +2680*(4*n-15)*(2*n-7)*(4*n-13)*a(n-4)=0. - R. J. Mathar, Oct 19 2021
From Vaclav Kotesovec, Nov 01 2021: (Start)
Recurrence (of order 2): 16*(n-1)*n*(2*n - 1)*(51*n^2 - 162*n + 127)*a(n) = (n-1)*(5457*n^4 - 22791*n^3 + 32144*n^2 - 17536*n + 3072)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(51*n^2 - 60*n + 16)*a(n-2).
a(n) ~ sqrt(3 + 5/sqrt(17)) * (107 + 51*sqrt(17))^n / (sqrt(Pi*n) * 2^(6*n+2)). (End)
From Peter Bala, Feb 21 2022: (Start)
a(n) = [x^n] ( (1 - x)*(1 - x^2) )^(-n). Cf. A234839.
a(n) = Sum_{k = 0..floor(n/2)} binomial(2*n-2*k-1,n-2*k)*binomial(n+k-1,k).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 9*x^3 + 32*x^4 + 119*x^5 + ... is the g.f. of A063020.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 5 and positive integers n and k.
The o.g.f. A(x) is the diagonal of the bivariate rational function 1/(1 - t/((1-x)*(1-x^2))) and hence is an algebraic function over Q(x) by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^2) ). Then A(x) = 1 + x*d/dx (log(F(x))). (End)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(2*n+k-1, k)*binomial(2*n-k-1, n-k). Cf. A352373. - Peter Bala, Jun 05 2024

More terms from Alois P. Heinz, Oct 17 2021

A133675 Negative discriminants with form class number 1 (negated).

Original entry on oeis.org

3, 4, 7, 8, 11, 12, 16, 19, 27, 28, 43, 67, 163

Offset: 1

N. J. A. Sloane, May 16 2003

The list on p. 260 of Cox is missing -12, the list in Theorem 7.30 on p. 149 is correct. - Andrew V. Sutherland, Sep 02 2012

Let b(k) be the number of integer solutions of f(x,y) = k, where f(x,y) is the principal binary quadratic form with discriminant d<0 (i.e., f(x,y) = x^2 - (d/4)*y^2 if 4|d, x^2 + x*y + ((1-d)/4)*y^2 otherwise), then this sequence lists |d| such that {b(k)/b(1): k>=1} is multiplicative. See Crossrefs for the actual sequences. - Jianing Song, Nov 20 2019

D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, pp. 149, 260.
D. E. Flath, Introduction to Number Theory, Wiley-Interscience, 1989.

Cf. A014602, A003173.
The sequences {b(k): k>=0}: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A033716 (d=-12), A004531 (d=-16), A028641 (d=-19), A138805 (d=-27), A033719 (d=-28), A138811 (d=-43), A318984 (d=-67), A318985 (d=-163).
The sequences {b(k)/b(1): k>=1}: A002324 (d=-3), A002654 (d=-4), A035182 (d=-7), A002325 (d=-8), A035179 (d=-11), A096936 (d=-12), A113406 (d=-16), A035171 (d=-19), A138806 (d=-27), A110399 (d=-28), A035147 (d=-43), A318982 (d=-67), A318983 (d=-163).

ok(n)={(-n)%4<2 && quadclassunit(-n).no == 1} \\ Andrew Howroyd, Jul 20 2018

Corrected by David Brink, Dec 29 2007

A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jul 15 2004

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

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).

Antti Karttunen, Table of n, a(n) for n = 1..65537
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.

Cf. A033716, A093766, A115979.

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then  a := a+1 ; end if; end do: a; end proc:
A002324 := proc(n) sigmamr(n,3,1)-sigmamr(n,3,2) ; end proc:
A096936 := proc(n) A002324(n) +2*(sigmamr(n,12,4)-sigmamr(n,12,8) ); end proc:
seq(A096936(n),n=1..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1 || # == 3, 1, # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 1, #2 + 1, True, (1 + (-1)^#2)/2] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 20 2017 *)

{a(n) = if( n<1, 0, 1/2 * polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n)) * sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x*O(x^n)), n))};

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

{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==3, 1, p==2, 3 * (1 + (-1)^e) / 2, p%3==2, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Nov 20 2017 */

(definec (A096936 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n)) (rest (A096936 (A028234 n)))) (cond ((= 2 p) (* (if (odd? e) 0 3) rest)) ((= 3 p) rest) ((= 1 (modulo p 3)) (* (+ 1 e) rest)) (else (* (if (odd? e) 0 1) rest)))))) ;; With the memoization-macro definec, after the given multiplicative formula. - Antti Karttunen, Nov 20 2017

a(n) = A033716(n) / 2.
Multiplicative with a(2^e) = 3*(1+(-1)^e)/2, a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if p==2 (mod 3) and a(p^e) = 1+e if p==1 (mod 3). - Corrected by Antti Karttunen, Nov 20 2017
G.f.: ((Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(3*k^2)) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Oct 15 2022

A320078 Expansion of Product_{k>0} theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 0, 2, 6, 2, 4, 6, 8, 16, 8, 14, 26, 26, 24, 30, 58, 50, 60, 78, 90, 118, 104, 138, 192, 224, 204, 268, 366, 354, 412, 474, 596, 694, 724, 818, 1052, 1162, 1176, 1470, 1756, 1918, 2052, 2434, 2814, 3168, 3396, 3806, 4674, 5124, 5396, 6250, 7374, 7898, 8732

Offset: 0

Seiichi Manyama, Oct 05 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000122, A033716, A320067, A320098.

nmax = 60; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 60; CoefficientList[Series[Product[(1 - x^((2*k-1)*j))*(1 + x^((2*k-1)*j))^3/(1 + x^(2*j*(2*k-1)))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*(2*k-1)))^5/( eta(q^(2*k-1))* eta(q^(4*(2*k-1))))^2 ) ) \\ G. C. Greubel, Oct 29 2018

Expansion of Product_{k>0} eta(q^(2*(2*k-1)))^5 / (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2.

a(n) ~ (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 07 2018

A034896 Number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.

Original entry on oeis.org

1, 4, 4, 4, 20, 24, 4, 32, 52, 4, 24, 48, 20, 56, 32, 24, 116, 72, 4, 80, 120, 32, 48, 96, 52, 124, 56, 4, 160, 120, 24, 128, 244, 48, 72, 192, 20, 152, 80, 56, 312, 168, 32, 176, 240, 24, 96, 192, 116, 228, 124, 72, 280, 216, 4, 288, 416, 80, 120, 240, 120, 248, 128, 32, 500

Offset: 0

N. J. A. Sloane

nonn

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

Number 16 of the 126 eta-quotients listed in Table 1 of Williams 2012. - Michael Somos, Nov 10 2018

			G.f. = 1 + 4*x + 4*x^2 + 4*x^3 + 20*x^4 + 24*x^5 + 4*x^6 + 32*x^7 + ... - _Michael Somos_, Nov 10 2018

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Gilleland, Some Self-Similar Integer Sequences
J. Liouville, Sur la forme x^2 + y^2 + 3(z^2 + t^2), Journal de mathématiques pures et appliquées 2e série, tome 5 (1860), p. 147-152.
K. S. Williams, Fourier series of a class of eta quotients, Int. J. Number Theory 8 (2012), no. 4, 993-1004.

Cf. A272364, A320147, A320148.

Cf. A033716, A113262, A134946.

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* Michael Somos, Nov 10 2018 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^10 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A))^4, n))}; /* Michael Somos, Nov 10 2018 */

Expansion of theta_3(q)^2*theta_3(q^3)^2.
G.f.: s(2)^10*s(6)^10/(s(1)*s(3)*s(4)*s(12))^4, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
Fine gives an explicit formula for a(n) in terms of the divisors of n.
From Michael Somos, Nov 10 2018: (Start)
Expansion of (a(q) + 2*a(q^4))^2 / 9 = (a(q)^2 - 2*a(q^2)^2 + 4*a(q^4)^2) / 3 in powers of q where a() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + 4 Sum_{k>0} k x^k / (1 - (-x)^k) Kronecker(9, k).
a(n) = 1 + 4 * A113262(n) = (-1)^n * A134946(n). Convolution square of A033716.
a(n) = 4 * (s(n) - 2*s(n/2) - 3*s(n/3) + 4*s(n/4) + 6*s(n/6) - 12*s(n/12)) if n>0 where s(x) = sum of divisors of x for integer x else 0. (End)

A119395 Number of nonnegative integer solutions to the equation x^2 + 3y^2 = n.

Original entry on oeis.org

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

Offset: 0

Max Alekseyev, May 16 2006

nonn

The number of integer solutions is given by A033716.

Records 1, 2, 3, 5, 6, 9, 12, 14, 18, ... occur at 0, 4, 28, 196, 364, 2548, 6916, 33124, 48412, ... - Antti Karttunen, Nov 20 2017

Antti Karttunen, Table of n, a(n) for n = 0..65537
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A033716, A096936.

QP = QPochhammer;
s = (QP[q^2]*QP[q^6])^5/(QP[q]*QP[q^3]*QP[q^4]*QP[q^12])^2 + O[q]^105;
A033716 = CoefficientList[s, q];
A119395 = Ceiling[A033716/4] (* Jean-François Alcover, Jul 02 2018 *)

{ A033716(n) = local(f,B); f=factorint(n); B=1; for(i=1,matsize(f)[1], if(f[i,1]%3==1,B*=f[i,2]+1); if(f[i,1]%3==2,if(f[i,2]%2,return(0)))); if(n%4,2*B,6*B) } { a(n) = ceil(A033716(n)/4) }

For n > 0, a(n) = (A033716(n) + 2)/4 if n is a square or a triple of a square; otherwise a(n) = A033716(n)/4. Alternatively, a(n) = ceiling(A033716(n)/4).

G.f.: (1 + theta_3(q))*(1 + theta_3(q^3))/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, -2, 6, 0, 0, -4, 0, -2, 0, 0, 6, -4, 0, 0, 6, 0, 0, -4, 0, -4, 0, 0, 0, -2, 0, -2, 12, 0, 0, -4, 0, 0, 0, 0, 6, -4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, -6, 0, 0, 12, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 12, 0, 0, -4, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, -4, 0, 0, 6, 0

Offset: 0

Michael Somos, Feb 09 2006

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

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. A004016, A033716, A033762, A097195, A112604, A122861.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Nov 09 2013 *)

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

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==2, -3 * ((e+1)%2), if( p==3, 1, if( p%6==1, e+1, (e+1)%2))))))} /* Michael Somos, Nov 09 2013 */

Expansion of theta_4(q) * theta_4(q^3) in powers of q.
Expansion of (4 * a(q^4) - a(q)) / 3 = (4 * b(q^4) - b(q)) * b(q) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Nov 09 2013
Expansion of (eta(q) * eta(q^3))^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -2, -1, -4, -1, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 2, 0, 6, 2, 0, -2, -6, 0, -2, 2, 0, ...]. - Michael Somos, Nov 09 2013
a(n) = -2*b(n) where b(n) is multiplicative and b(2^e) = -3 * (1 + (-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = 1+e if p == 1 (mod 6), b(p^e) = (1 +(-1)^e) / 2 if p == 5 (mod 6).
Given g.f. A(x), then B(x) = A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v*(u + v)^2 - 4*u * (w^2 - v*w + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033762. - Michael Somos, Nov 09 2013
G.f.: 1 - 2*(Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - 4 * x^(4*k) / (1 + x^(4*k) + x^(8*k))).
G.f.: (Sum_{k in Z} (-x)^(k^2)) * (Sum_{k in Z} (-x)^(3*k^2)).
a(n) = -2 * A115979(n) unless n=0. a(n) = (-1)^n * A033716(n).
a(3*n + 2) = a(4*n + 2) = 0. a(3*n) = a(n). a(2*n + 1) = -2 * A033762(n). a(3*n + 1) = -2 * A122861(n). a(4*n) = A004016(n). a(4*n + 1) = -2 * A112604(n). a(6*n + 1) = -2 * A097195(n). - Michael Somos, Nov 09 2013

cp's OEIS Frontend

A004016 Theta series of planar hexagonal lattice A_2.

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A025428 Number of partitions of n into 4 nonzero squares.

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A092573 Number of solutions to x^2 + 3y^2 = n in positive integers x and y.

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A348410 Number of nonnegative integer solutions to n = Sum_{i=1..n} (a_i + b_i), with b_i even.

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A133675 Negative discriminants with form class number 1 (negated).

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A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

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A034896 Number of solutions to a^2 + b^2 + 3c^2 + 3d^2 = n.