A002324 -id:A002324 - cp's OEIS frontend

A001817 G.f.: Sum_{n>0} x^n/(1-x^(3n)) = Sum_{n>=0} x^(3n+1)/(1-x^(3n+1)).

1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 3, 2, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 2, 1, 3, 3, 3, 1, 4, 1, 1, 2, 4, 2, 2, 1, 3, 2, 2, 2, 4, 2, 2, 2, 3, 1, 4, 1, 2, 2, 2, 2, 4, 2, 2, 2, 5, 1, 2, 1, 4, 2, 2, 1, 4, 1, 2, 4, 3, 2, 2, 2, 3, 2, 3, 1, 5, 1, 2, 2, 4, 2

Offset: 1

N. J. A. Sloane

a(n) is the number of positive divisors of n of the form 3k+1. If r(n) denotes the number of representations of n by the quadratic form j^2+ij+i^2, then r(n)= 6 *(a(n)-A001822(n)). - Benoit Cloitre, Jun 24 2002

			x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + x^9 + ...

Bruce C. Berndt, On a certain theta-function in a letter of Ramanujan from Fitzroy House, Ganita 43 (1992), 33-43.

Nick Hobson, Table of n, a(n) for n = 1..10000
P. G. Dirichlet, Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 21 (1840), 1-12.
Michael Gilleland, Some Self-Similar Integer Sequences.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A001822, A035191, A002324.
Cf. A051731.
Cf. A001620, A256425.

a001817 n = length [d | d <- [1,4..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 26 2011

A001817 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) = 1 then
            a := a+1 ;
        end if ;
    end do:
    a ;
end proc:
seq(A001817(n),n=1..100) ; # R. J. Mathar, Sep 25 2017

a[n_] := DivisorSum[n, Boole[Mod[#, 3] == 1]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

PARI
```
a(n)=if(n<1,0,sumdiv(n,d,d%3==1))
```

Moebius transform is period 3 sequence [1, 0, 0, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^(3k-2)/(1-x^(3k-2)) = Sum_{k>0} x^k/(1-x^(3k)). - Michael Somos, Sep 20 2005
Equals A051731 * [1, 0, 0, 1, 0, 0, 1, 0, 0, 1, ...]. - Gary W. Adamson, Nov 06 2007
a(n) = (A035191(n) + A002324(n)) / 2. - Reinhard Zumkeller, Nov 26 2011
Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,3) - (1 - gamma)/3 = A256425 - (1 - A001620)/3 = 0.536879... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 33.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant 33. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Amiram Eldar, Table of n, a(n) for n = 1..10000

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. A038907, A038908.

a[n_] := DivisorSum[n, KroneckerSymbol[33, #] &]; Array[a, 100] (* Amiram Eldar, Nov 19 2023 *)

my(m = 33); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

a(n) = sumdiv(n, d, kronecker(33, d)); \\ Amiram Eldar, Nov 19 2023

From Amiram Eldar, Nov 19 2023: (Start)
a(n) = Sum_{d|n} Kronecker(33, d).
Multiplicative with a(p^e) = 1 if Kronecker(33, p) = 0 (p = 3 or 11), a(p^e) = (1+(-1)^e)/2 if Kronecker(33, p) = -1 (p is in A038908), and a(p^e) = e+1 if Kronecker(33, p) = 1 (p is in A038907 \ {3, 11}).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*log(4*sqrt(33)+23)/sqrt(33) = 1.332797188186... . (End)

A001822 Expansion of Sum_{n>=0} x^(3n+2)/(1-x^(3n+2)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

a(n) is the number of positive divisors of n of the form 3k+2. If r(n) denotes the number of representations of n by the quadratic form j^2+i*j+i^2, then r(n)= 6 *(A001817(n)-a(n)). - Benoit Cloitre, Jun 24 2002

Bruce C. Berndt,"On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992),33-43.

Nick Hobson, Table of n, a(n) for n = 1..10000
P. G. Dirichlet, Recherches sur diverses applications de l'analyse infinitésimale à la théorie des nombres, J. Reine Angew. Math. 21 (1840), 1-12.
Michael Gilleland, Some Self-Similar Integer Sequences.
R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Cf. A000005, A001620, A001817, A002324, A035191, A256843.

a001822 n = length [d | d <- [2,5..n], mod n d == 0]
-- Reinhard Zumkeller, Nov 26 2011

A001822 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if modp(d,3) = 2 then
            a := a+1 ;
        end if ;
    end do:
    a ;
end proc:
seq(A001822(n),n=1..100) ; # R. J. Mathar, Sep 25 2017

a[n_] := DivisorSum[n, Boole[Mod[#, 3] == 2]&]; Array[a, 100] (* Jean-François Alcover, Dec 01 2015 *)

PARI
```
a(n)=if(n<1, 0, sumdiv(n,d, d%3==2))
```

Moebius transform is period 3 sequence [0, 1, 0, ...]. - Michael Somos, Sep 20 2005
G.f.: Sum_{k>0} x^(3k-1)/(1-x^(3k-1)) = Sum_{k>0} x^(2k)/(1-x^(3k)). - Michael Somos, Sep 20 2005
a(n) = (A035191(n) - A002324(n)) / 2. - Reinhard Zumkeller, Nov 26 2011
a(n) + A001817(n) + A000005(n/3) = A000005(n), where A000005(.)=0 if the argument is not an integer. - R. J. Mathar, Sep 25 2017
Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(2,3) - (1 - gamma)/3 = A256843 - (1 - A001620)/3 = -0.0677207... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

Original entry on oeis.org

1, 2, 2, 4, 3, 5, 3, 7, 5, 7, 4, 11, 5, 8, 8, 12, 6, 13, 6, 15, 10, 11, 7, 21, 10, 13, 12, 18, 9, 22, 9, 21, 14, 16, 14, 29, 11, 17, 16, 29, 12, 28, 12, 25, 23, 20, 13, 39, 16, 27, 20, 29, 15, 34, 20, 36, 22, 25, 16, 50, 17, 26, 29, 38, 24, 40, 18, 36, 26, 40

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

From Andrey Zabolotskiy, Mar 12 2018: (Start)
If reflections are not allowed, we get A145392. If any rotations and reflections are allowed, we get A054346.
The parent lattice of the sublattices under consideration has Patterson symmetry group p4mm, and two sublattices are considered equivalent if they are related via a symmetry from that group [Rutherford]. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145391 (c2mm), A145392 (p4), A145394 (p6), A003051 (p6mm).
Rutherford says at p. 161 that a(n) != A054346(n) only when A002654(n) > 2, but actually these two sequence differ at other terms, too, for example, at n = 30 (see illustration). (End)

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see table 6 and fig. 2).
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 2; beware the typo in a(5).]
Andrey Zabolotskiy, Sublattices of the square lattice (illustrations for n = 1..6, 15, 25)
Index entries for sequences related to sublattices
Index entries for sequences related to square lattice

Cf. A054345, A054346, A167156, A008621.
Cf. A000203, A069734, A145391, A145392, A145394, A003051, A002324, A002654, A069735, A145390.
Cf. A019590, A157228, A157226, A157230, A157231, A053866, A025441.

terms = 70;
CoefficientList[Sum[(1/((1-x^m)(1-x^(4m)))-1), {m, 1, terms}] + O[x]^(terms + 1), x] // Rest (* Jean-François Alcover, Aug 05 2018 *)

a(n) = (A000203(n) + A002654(n) + A069735(n) + A145390(n))/4. [Rutherford] - N. J. A. Sloane, Mar 13 2009
G.f.: Sum_{ m>=1 } (1/((1-x^m)(1-x^(4m))) - 1). [Hanany, Orlando & Reffert, eq. (6.8)] - Andrey Zabolotskiy, Jul 05 2017
a(n) = Sum_{ m: m^2|n } A019590(n/m^2) + A157228(n/m^2) + A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2) = A053866(n) + A025441(n) + Sum_{ m: m^2|n } A157226(n/m^2) + A157230(n/m^2) + A157231(n/m^2). [Rutherford] - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A008621(d) = Sum_{ d|n } (1 + floor(d/4)). [From the above-given g.f.] - Andrey Zabolotskiy, Jul 17 2019

New name from Andrey Zabolotskiy, Mar 12 2018

A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Nov 16 2003

nonn

Also, apparently the number of 6-regular plane graphs with n vertices that have only trigonal faces and loops ("({1,3},6)-spheres" from the paper by Michel Deza and Mathieu Dutour Sikiric). - Andrey Zabolotskiy, Dec 22 2021

			From _M. F. Hasler_, Mar 05 2018: (Start)
a(0) = a(1) = 1 since 0 = 0^2 + 0*0 + 0^2 and 1 = 0^2 + 0*1 + 1^2.
a(2) = 0 since 2 cannot be written as x^2 + xy + y^2.
a(49) = 2 since 49 = 0^2 + 0*7 + 7^2 = 3^2 + 3*5 + 5^2. (End)

B. C. Berndt, "On a certain theta-function in a letter of Ramanujan from Fitzroy House", Ganita 43 (1992), 33-43.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Michel Deza and Mathieu Dutour Sikiric, Zigzag and Central Circuit Structure of ({1,2,3},6)-spheres, Taiwanese Journal of Mathematics, 16 (June 2012), No. 3, 913-940; see Table 1, p. 916.
Michel-Marie Deza, Mathieu Dutour Sikiric, and Mikhail Ivanovitch Shtogrin, Geometric Structure of Chemistry-Relevant Graphs, Springer, 2015; see Table 5.4 and Section 5.4.
Oscar Marmon, Hexagonal Lattice Points on Circles, arXiv:math/0508201 [math.NT], 2005.

Cf. A003136, A034020, A000196.
Cf. A118886 (indices of values > 1), A198772 (indices of 1's), A198773 (indices of 2's), A198774 (indices of 3's), A198775 (indices of 4's), A198799 (index of 1st term = n).
Cf. A215622.

a088534 n = length
   [(x,y) | y <- [0..a000196 n], x <- [0..y], x^2 + x*y + y^2 == n]
a088534_list = map a088534 [0..]
-- Reinhard Zumkeller, Oct 30 2011

function A088534(n)
    n % 3 == 2 && return 0
    M = Int(round(2*sqrt(n/3)))
    count = 0
    for y in 0:M, x in 0:y
        n == x^2 + y^2 + x*y && (count += 1)
    end
    return count
end
A088534list(upto) = [A088534(n) for n in 0:upto]
A088534list(104) |> println # Peter Luschny, Mar 17 2018

a[n_] := Sum[Boole[i^2 + i*j + j^2 == n], {i, 0, n}, {j, 0, i}];
Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Jun 20 2018 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+i*j+j^2-n,0,1)))

A088534(n,d)=sum(x=0,sqrt(n\3),sum(y=max(x,sqrtint(n-x^2)\2),sqrtint(n-2*x^2),x^2+x*y+y^2==n&&(!d||!printf("%d",[x,y]))))\\ Set 2nd arg = 1 to print all decompositions, with 0 <= x <= y. - M. F. Hasler, Mar 05 2018

def A088534(n):
    c = 0
    for y in range(n+1):
        if y**2 > n:
            break
        for x in range(y+1):
            z = x*(x+y)+y**2
            if z > n:
                break
            elif z == n:
                c += 1
    return c # Chai Wah Wu, May 16 2022

a(A003136(n)) > 0; a(A034020(n)) = 0;
a(A118886(n)) > 1; a(A198772(n)) = 1;
a(A198773(n)) = 2; a(A198774(n)) = 3;
a(A198775(n)) = 4;
a(A198799(n)) = n and a(m) <> n for m < A198799(n). - Reinhard Zumkeller, Oct 30 2011, corrected by M. F. Hasler, Mar 05 2018
In the prime factorization of n, let S_1 be the set of distinct prime factors p_i for which p_i == 1 (mod 3), let S_2 be the set of distinct prime factors p_j for which p_j == 2 (mod 3), and let M be the exponent of 3. Then n = 3^M * (Product_{p_i in S_1} p_i ^ e_i) * (Product_{p_j in S_2} p_j ^ e_j), and the number of solutions for x^2 + xy + y^2 = n, 0 <= x <= y is floor((Product_{p_i in S_1} (e_i + 1) + 1) / 2) if all e_j are even and 0 otherwise. E.g. a(1729) = 4 since 1729 = 7^1*13^1*19^1 and floor(((1+1)*(1+1)*(1+1)+1)/2) = 4. - Seth A. Troisi, Jul 02 2020
a(n) = ceiling(A004016(n)/12) = (A002324(n) + A145377(n)) / 2. - Andrey Zabolotskiy, Dec 23 2021

Edited by M. F. Hasler, Mar 05 2018

A074941 a(n) = sigma(n) 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, 0, 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, 1, 0, 2, 0, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0

Offset: 1

Benoit Cloitre, Oct 04 2002

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A010872.

Differs from A002324 for the first time at n=49, where a(49) = 0, while A002324(49) = 3.

A074941:= n-> (numtheory[sigma](n) mod 3):
seq (A074941(n), n=1..105); # Jani Melik, Jan 26 2011

a[n_] := Mod[DivisorSigma[1, n], 3]; Array[a, 100] (* Amiram Eldar, Jun 06 2022 *)

PARI
```
a(n)=sigma(n)%3
```

a(n) = A010872(A000203(n)). - Antti Karttunen, Nov 05 2017

A112608 Number of representations of n as a sum of a twice a square and three times a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

The greedy inverse (first occurrence of n) starts 1, 0, 2, 18, 11, 900, 116, 44118, 515, 3105, 5702, ... - R. J. Mathar, Apr 28 2020

			a(11) = 4 since we can write 11 = 2*(2)^2 + 3*1 = 2*(-2)^2 + 3*1 = 2*(1)^2 + 3*3 = 2*(-1)^2 + 3*3

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

eta[x_] := x^(1/24)*QPochhammer[x]; A112608[n_] := SeriesCoefficient[ q^(-3/8)*(eta[q^4]^5*eta[q^6]^2)/(eta[q^2]^2*eta[q^3]*eta[q^8]^2), {q, 0, n}]; Table[A112608[n], {n, 0, 50}] (* G. C. Greubel, Sep 25 2017 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^4+A)^5*eta(x^6+A)^2/ eta(x^2+A)^2/eta(x^3+A)/eta(x^8)^2, n))} /* Michael Somos, Jan 01 2006 */

a(n) = d_{1, 3}(8n+3) - d_{2, 3}(8n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Euler transform of period 24 sequence [0, 2, 1, -3, 0, 1, 0, -1, 1, 2, 0, -4, 0, 2, 1, -1, 0, 1, 0, -3, 1, 2, 0, -2, ...]. - Michael Somos, Jan 01 2006
Expansion of q^(-3/8)*(eta(q^4)^5*eta(q^6)^2)/(eta(q^2)^2*eta(q^3)*eta(q^8)^2) in powers of q.
a(n) = A002324(8*n+3).

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

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(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.

A230655 Squared radii of circles around a point of the hexagonal lattice that contain a record number of lattice points.

Original entry on oeis.org

0, 1, 7, 49, 91, 637, 1729, 8281, 12103, 53599, 157339, 375193, 1983163, 4877509, 13882141, 85276009, 180467833, 596932063, 3428888827, 4178524441, 7760116819, 29249671087, 36412855843, 147442219561, 254889990901, 473367125959, 1784229936307, 2439661341481

Offset: 1

Hugo Pfoertner, Oct 27 2013

nonn

It appears that this is also the sequence of numbers with a record number of divisors all of whose prime factors are of the form 3k + 1. - Amiram Eldar, Sep 12 2019 [This is correct, see A343771. - Jianing Song, May 19 2021]

Indices of records of A004016. Apart from the first term, also indices of records of A002324. - Jianing Song, May 20 2021

			a(2)=7 because a circle with radius sqrt(7) around the lattice point at (0,0) is the first circle that passes through more lattice points than a circle with radius 1, which passes through 6 points. The 12 hit points are (+-1/2,+-3*sqrt(3)/2), (+-2,+-sqrt(3)), (+-5/2, +-sqrt(3)/2).

Jianing Song, Table of n, a(n) for n = 1..247 (all terms <= 10^75.)

Cf. A004016, A002324.
Cf. A003136 (all occurring squared radii), A198799 (common terms), A230656 (index positions of records), A344472 (records).
Apart from the first term, subsequence of A343771.
Indices of records of Sum_{d|n} kronecker(m, d): this sequence (m=-3), A071383 (m=-4, similar sequence for square lattice), A279541 (m=-6).

my(v=list_A344473(10^15), rec=0); print1(0, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(v[n], ", "))) \\ Jianing Song, May 20 2021, see program for A344473

Offset corrected by Jianing Song, May 20 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

cp's OEIS Frontend

A001817 G.f.: Sum_{n>0} x^n/(1-x^(3n)) = Sum_{n>=0} x^(3n+1)/(1-x^(3n+1)).

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A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = 33.

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A001822 Expansion of Sum_{n>=0} x^(3n+2)/(1-x^(3n+2)).

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A145393 Number of inequivalent sublattices of index n in square lattice, where two sublattices are considered equivalent if one can be rotated or reflected to give the other, with that rotation or reflection preserving the parent square lattice.

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A088534 Number of representations of n by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

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A074941 a(n) = sigma(n) mod 3.

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A035215 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = 33.