A002652 -id:A002652 - cp's OEIS frontend

A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618.

1, 2, 22, 336, 6006, 117348, 2428272, 52303680, 1160427510, 26337699740, 608642155660, 14272471122560, 338764038330480, 8123136091556640, 196484811079765440, 4788469475873867520, 117465323079289162230, 2898183118626011393100

Offset: 1

Lynette O'Brien, Dec 15 2016

nonn

G.f. is the square root of the g.f. for A183204.
This sequence is c_n in Theorem 6.1 in O'Brien's thesis.
Also see Conjecture 5.4 in Chan, Cooper and Sica's paper.

			G.f. = 1 + 2*x + 22*x^2 + 336*x^3 + 6006*x^4 + ....

L. O'Brien, Modular forms and two new integer sequences at level 7, Massey University, 2016.

G. C. Greubel, Table of n, a(n) for n = 1..500
H. H. Chan, S. Cooper, F. Sica, Congruences satisfied by Apéry-like numbers, International Journal of Number Theory, 2010, 6(01), 89-97. Conjecture 5.4.
Lynette O'Brien, Modular forms and two new integer sequences at level 7
Lynette O'Brien, Modular forms and two new integer sequences at level 7

Cf. A183204, A279613, A279618.

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

I:=[2, 22]; [1] cat [n le 2 select I[n] else ((26*n^2-39*n+15)* Self(n-1) + 3*(3*n-4)*(3*n-5)*Self(n-2))/n^2 : n in [1..50]] // G. C. Greubel, Jul 04 2018

RecurrenceTable[{a[n+1] == ((26*n^2+13*n+2)*a[n] + 3*(3*n-1)*(3*n-2)*a[n-1])/ (n + 1)^2, a[-1] == 0, a[0] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jul 04 2018 *)
CoefficientList[Series[Sqrt[7]*(1/(25 - 80*x + 24*Sqrt[1 - 27*x]*Sqrt[1+x]))^(1/4) * Hypergeometric2F1[1/12, 5/12, 1, 13824*x^7/(1 - 21*x + 8*x^2 + Sqrt[1 - 27*x] * (1 - 8*x)*Sqrt[1+x])^3], {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 04 2018 *)

(n+1)^2*a_7(n+1) = (26*n^2+13*n+2)*a_7(n) + 3*(3*n-1)*(3*n-2)*a_7(n-1), a(0)=1, a(-1)=0.
Conjecture: For any positive integer n and any prime p with p equiv. 0,1,2 or 4 modulo 7, a(n) equiv. a(n)=a(n_0)a(n_1)...a(n_r) modulo p, where n=n_0+n_1p+...n_rp^r is the base p representation of n.
Conjecture: a(n)~ C n^(-3/2) 27^n where C=0.0955223052681267146513079107870296256727946666510071798669948234917659...

A028951 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 4 ] (or the Kleinian 2-d lattice, see A002652).

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 9, 11, 14, 16, 18, 22, 23, 25, 28, 29, 32, 36, 37, 43, 44, 46, 49, 50, 53, 56, 58, 63, 64, 67, 71, 72, 74, 77, 79, 81, 86, 88, 92, 98, 99, 100, 106, 107, 109, 112, 113, 116, 121, 126, 127, 128, 134, 137, 142, 144, 148, 149, 151, 154, 158, 161, 162

Offset: 1

N. J. A. Sloane

nonn

Or, numbers of the form x^2+xy+2y^2 with x and y integers. - N. J. A. Sloane, Apr 30 2015

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A028929, A035248, A002652, A034036, A257346 (complement).

Reap[For[n = 0, n < 200, n++, r = Reduce[x^2 + x y + 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

More terms from Larry Reeves (larryr(AT)acm.org), Mar 29 2000

A205975 a(n) = Fibonacci(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

Original entry on oeis.org

1, 2, 4, 0, 18, 0, 0, 26, 168, 68, 0, 356, 0, 0, 1508, 0, 9870, 0, 10336, 0, 0, 0, 141688, 114628, 0, 150050, 0, 0, 1906866, 2056916, 0, 0, 26139708, 0, 0, 0, 89582112, 96631268, 0, 0, 0, 0, 0, 1733977748, 8416904796, 0, 14690495224, 0, 0, 15557484098

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Compare the g.f. to the Lambert series of A002652:

1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 18*x^4 + 26*x^7 + 168*x^8 + 68*x^9 + 356*x^11 +...
where A(x) = 1 + 1*2*x + 1*4*x^2 + 3*6*x^4 + 14*2*x^7 + 21*8*x^8 + 34*2*x^9 + 89*4*x^11 + 377*4*x^14 + 987*10*x^16 +...+ Fibonacci(n)*A002652(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1-x-x^2) + 1*x^2/(1-3*x^2+x^4) - 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) - 5*x^5/(1-11*x^5-x^10) - 8*x^6/(1-18*x^6+x^12) + 0*13*x^7/(1+29*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

Cf. A002652, A205974, A205976, A203847, A000204 (Lucas).

Cf. A209455 (Pell variant).

terms = 50; s = 1 + 2 Sum[Fibonacci[n]*KroneckerSymbol[n, 7]*x^n/(1 - LucasL[n]*x^n + (-1)^n*x^(2*n)), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)

{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1 + 2*sum(m=1,n,fibonacci(m)*kronecker(m,7)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

G.f.: 1 + 2*Sum_{n>=1} Fibonacci(n)*Kronecker(n,7)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)).

A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

Original entry on oeis.org

1, 2, 8, 0, 72, 0, 0, 338, 3264, 1970, 0, 22964, 0, 0, 323128, 0, 4708320, 0, 10976840, 0, 0, 0, 745778864, 900234724, 0, 2623476242, 0, 0, 110745336312, 178241928596, 0, 0, 7524162792576, 0, 0, 0, 127800022137480, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A002652: 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 72*x^4 + 338*x^7 + 3264*x^8 + 1970*x^9 +...
where A(x) = 1 + 1*2*x + 2*4*x^2 + 12*6*x^4 + 169*2*x^7 + 408*8*x^8 + 985*2*x^9 + 5741*4*x^11 + 80782*4*x^14 + 470832*10*x^16 +...+ Pell(n)*A002652(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 2*( 1*x/(1-2*x-x^2) + 2*x^2/(1-6*x^2+x^4) - 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002652, A205975, A209454, A209456, A204270, A000129 (Pell), A002203.

terms = 44; s = 1 + 2 Sum[x^n*Fibonacci[n, 2]*KroneckerSymbol[n, 7]/(1 + (-1)^n*x^(2*n) - x^n*(Fibonacci[n - 1, 2] + Fibonacci[n + 1, 2])), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
A002652[n_]:= If[n < 1, Boole[n == 0], 2*Sum[KroneckerSymbol[-7, d], {d, Divisors[n]}]]; Join[{1}, Table[Fibonacci[n, 2]*A002652[n], {n,1,50}]] (* G. C. Greubel, Jan 03 2017 *)

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(1 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

G.f.: 1 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

G.f.: 1 + 2*Sum_{n>=1} F(n,2)*Kronecker(n,7)*x^n/(1 + (-1)^n*x^(2*n)-x^n* (F(n-1,2)+F(n+1,2))), where F is the Fibonacci polynomial. - Jean-François Alcover, Jul 05 2017

A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0, 0, 4, 0, 4, 2, 0, 8, 0, 0, 0, 0, 2, 8, 4, 0, 6, 0, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 4, 2, 2, 8, 0, 0, 8, 0, 0, 8, 0, 4, 0, 0, 0, 0, 2, 0, 8, 4, 4, 0, 0, 0, 6, 4, 0, 4, 4, 0, 0, 0, 0, 10, 4, 4, 0, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 12, 2, 0, 8, 0

Offset: 0

N. J. A. Sloane

Theta series of lattice C2 with Gram matrix [ 1, 0; 0, 2]. a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Number 17 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by |a_4(n)| in Kassel and Reutenauer 2015. - Michael Somos, Jun 16 2015

			G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 2*q^4 + 4*q^6 + 2*q^8 + 6*q^9 + 4*q^11 + 4*q^12 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 114 Entry 8(iii).
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, 1999, p. 102, eq. 9.
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. 19.
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

John Cannon, Table of n, a(n) for n = 0..10000
George E. Andrews, Richard Lewis, and Zhi-Guo Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
Michael Gilleland, Some Self-Similar Integer Sequences.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
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.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852, Table I.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers, 2016.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002325, A002479, A028572, A033761, A082564, A112603, A113411, A133692, A139093, A226225, A226240, A242609, A245572, A247719.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), this sequence (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(8), 1), 105); A[1] + 2*A[2] + 2*A[3]; /* Michael Somos, Aug 29 2014 */

d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1,8,n)+d(3,8,n)-d(5,8,n)-d(7,8,n)),n=1..120)];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Sep 09 2012 *)
a[ n_] := If[ n < 1, Boole[ n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -2, #] &]]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ q^4])^3 / (QPochhammer[ q] QPochhammer[ q^8])^2, {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)

{a(n) = if( n<1, n==0, 2 * (issquare(n) - issquare(2*n) + 2 * sum( i=1, sqrtint(n\2), issquare(n - 2*i^2))))};

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -2, d)))}; /* Michael Somos, Aug 23 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 2], n)[n])}; /* Michael Somos, Aug 23 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n))};

Q = DiagonalQuadraticForm(ZZ, [1,2]); Q.representation_number_list(104); # Peter Luschny, Jun 20 2014

Fine gives an explicit formula for a(n) in terms of the divisors of n.
Euler transform of period 8 sequence [ 2, -1, 2, -4, 2, -1, 2, -2, ...].
Expansion of (eta(q^2) * eta(q^4))^3 / (eta(q) * eta(q^8))^2 in powers of q.
Coefficients in expansion of Sum_{i,j=-inf..inf} q^(i^2 + 2*j^2).
G.f. = s(2)^3*s(4)^3/(s(1)^2*s(8)^2), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-2, n) * x^k / (1 - x^k) = 1 + 2 * Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
G.f.: theta_3(q) * theta_3(q^2) = Product_{k>0} (1 + x^(2*k)) * ((1 + x^k) * (1 - x^(2*k)) / (1 + x^(4*k)))^2.
From Michael Somos, Oct 23 2006: (Start)
Moebius transform is period 8 sequence [ 2, 0, 2, 0, -2, 0, -2, 0, ...].
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 - 3*u3) * (u1 - u2 - u3 + u6) - (u2 - 3*u6) * (u1 - 2*u2 - u3 + 2*u6). (End)
a(n) = 2 * A002325(n) unless n = 0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 09 2012
From Michael Somos, Aug 29 2014: (Start)
Expansion of phi(q) * phi(q^2) in powers of q where phi() is a Ramanujan theta function.
a(2*n) = a(n). a(2*n + 1) = 2 * A113411(n). (End)
From Michael Somos, May 17 2015: (Start)
a(n) = A028572(4*n) = A133692(2*n) = A139093(8*n) = A226225(8*n) = A226240(4*n) = A242609(4*n) = A245572(4*n) / 3 = (-1)^floor((n + 1)/2) * A082564(n).
a(8*n + 5) = a(8*n + 7) = 0. a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = 4 * A033761(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(2^e) = 1, b(p^e) = e + 1 if p == 1, 3 (mod 8), b(p^e) = (1 + (-1)^e)/2 if p == 5, 7 (mod 8). - Jianing Song, Sep 04 2018 [Corrected by Jeremy Lovejoy, Nov 12 2024]
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi/sqrt(2) = 2.221441... (A247719). - Amiram Eldar, Dec 16 2023

A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 1, 4, 1, 0, 2, 0, 0, 2, 0, 5, 0, 2, 0, 0, 0, 4, 2, 0, 1, 0, 0, 3, 2, 0, 0, 6, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 2, 6, 0, 4, 0, 0, 1, 2, 0, 0, 2, 0, 0, 4, 0, 4, 0, 0, 0, 0, 1, 7, 0, 0, 2, 0, 0, 0, 2, 4, 0, 4, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 2, 3, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5*v^2 + 4*w^2 - 8*v*w - 4*u*v + 2*u*w + v - w. - Michael Somos, Jul 21 2004
Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - Michael Somos, Jun 05 2005
Inverse Moebius transform of A175629. - Jianing Song, Sep 07 2018
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -7. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + 2*x^2 + 3*x^4 + x^7 + 4*x^8 + x^9 + 2*x^11 + 2*x^14 + 5*x^16 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002652, A326919.
Moebius transform gives A175629.
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.

A := Basis( ModularForms( Gamma1(14), 1), 106); B := (-1 + A[1] + 2*A[2] + 4*A[3] + 6*A[5]) / 2; B; // Michael Somos, Jun 10 2015

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9] / 2]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; [ !(e%2), 1, e+1] [kronecker( -7, p) + 2]))}; \\ Michael Somos, May 28 2005

{a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -7, p)*X)))[n])}; \\ Michael Somos, Jun 05 2005

a(n) is multiplicative with a(7^e) = 1, a(p^e) = e + 1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, May 28 2005
2 * a(n) = A002652(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(7) =  1.187410... (A326919). - Amiram Eldar, Oct 11 2022

A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).

Original entry on oeis.org

1, 2, 0, 4, 2, 4, 0, 0, 0, 6, 0, 2, 4, 0, 0, 8, 2, 0, 0, 0, 4, 0, 0, 4, 0, 6, 0, 8, 0, 0, 0, 4, 0, 4, 0, 0, 6, 4, 0, 0, 0, 0, 0, 0, 2, 12, 0, 4, 4, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 4, 8, 0, 0, 0, 2, 0, 0, 4, 0, 8, 0, 4, 0, 0, 0, 12, 0, 0, 0, 0, 4, 10, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 0, 4, 0, 6, 6, 0, 0

Offset: 0

N. J. A. Sloane

Theta series of lattice with Gram matrix [2, 1; 1, 6].
Number of integer solutions (x, y) to x^2 + x*y + 3*y^2 = n. - Michael Somos, Sep 20 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^3 + 2*x^4 + 4*x^5 + 6*x^9 + 2*x^11 + 4*x^12 + 8*x^15 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 6] = 1 + 2*q^2 + 4*q^6 + 2*q^8 + 4*q^10 + 6*q^18 + 2*q^22 + 4*q^24 + 8*q^30 + 2*q^32 + 4*q^40 + 4*q^46 + 6*q^50 + 8*q^54 + 4*q^62 + 4*q^66 + 6*q^72 + 4*q^74 + ...

Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

John Cannon, Table of n, a(n) for n = 0..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A028610, A035179, A028954, A056874.
Cf. A000122, A000700, A010054, A121373.
Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), this sequence (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(11), 1), 103); A[1] + 2*A[2] + 4*A[4] + 2*A[5]; /* Michael Somos, Jul 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -11, #] &] 2]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^11] + EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^11], {q, 0, n}]; (* Michael Somos, Jul 12 2014 *)

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n * 4\11), 2 * issquare( t=4*n - 11*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * x * Ser(qfrep( [ 2, 1; 1, 6], n, 1)), n))}; /* Michael Somos, Apr 21 2015 */

{a(n) = if( n<1, n==0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -11, p) * X))[n] * 2)}; /* Michael Somos, Jun 05 2005 */

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -11, d)))}; /* Michael Somos, Jan 29 2007 */

Expansion of phi(x) * phi(x^11) = 4 * x^3 * psi(x^2) * psi(x^22) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 21 2015
From Michael Somos, Jan 29 2007: (Start)
Moebius transform is period 11 sequence [ 2, -2, 2, 2, 2, -2, -2, -2, 2, -2, 0, ...].
a(n) = 2 * b(n) and b(n) is multiplicative with b(11^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), b(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11).
G.f.: 1 + 2 * Sum_{k>0} Kronecker( -11, k) * x^k / (1 - x^k). (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^(1/2) (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 05 2007
Expansion of (F(x)^2 + 4 * F(x^2)^2 + 8 * F(x^4)^2) / F(x^2) in powers of x or expansion of (F(x)^2 + 2 * F(x^2)^2 + 2 * F(x^4)^2) / F(x^2) in powers of x^4 where F(x) = x^(1/2) * f(-x) * f(-x^11) and f() is a Ramanujan theta function. - Michael Somos, Mar 01 2010
a(n) = 2 * A035179(n) unless n=0. Convolution square is A028610.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(11) = 1.894451... . - Amiram Eldar, Dec 16 2023

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

A028641 Expansion of theta_3(q) * theta_3(q^19) + theta_2(q) * theta_2(q^19) in powers of q.

Original entry on oeis.org

1, 2, 0, 0, 2, 4, 0, 4, 0, 2, 0, 4, 0, 0, 0, 0, 2, 4, 0, 2, 4, 0, 0, 4, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 4, 0, 6, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 4, 0, 4, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 2, 8, 0, 0, 4, 2, 0, 4, 0, 8, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 4, 6, 4, 0, 0, 0

Offset: 0

N. J. A. Sloane

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

The number of integer solutions (x, y) to x^2 + x*y + 5*y^2 = n, discriminant -19. - Ray Chandler, Jul 12 2014

			G.f. = 1 + 2*x + 2*x^4 + 4*x^5 + 4*x^7 + 2*x^9 + 4*x^11 + 2*x^16 + 4*x^17 + 2*x^19 + ...

Robert Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 409, Eq. (19).

G. C. Greubel, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A035171.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), this sequence (d=-19), A138811 (d=-43).

a[ n_] := If[ n < 1, Boole[ n == 0], DivisorSum[ n, KroneckerSymbol[ -19, #] &] 2]; (* Michael Somos, Jun 14 2012 *)

{a(n) = if( n<1, n==0, sumdiv(n, d, kronecker(-19, d)) * 2)}; /* Michael Somos, Feb 27 2007 */

{a(n) = if( n<1, n==0, qfrep([2, 1;1, 10], n, 1)[n] * 2)}; /* Michael Somos, Feb 27 2007 */

Theta series of quadratic form with Gram matrix [ 2, 1; 1, 10 ].
Expansion of phi(q) * phi(q^19) + 4 * q^5 * psi(q^2)* psi(q^38) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 27 2007
Moebius transform is period 19 sequence [2, -2, -2, 2, 2, 2, 2, -2, 2, -2, 2, -2, -2, -2, -2, 2, 2, -2, 0, ...]. - Michael Somos, Feb 27 2007
a(n) = 2*b(n) where b(n) is multiplicative with a(0) = 1,  b(19^e) = 1, b(p^e) = e + 1 if Kronecker(-19, p) = 1, b(p^e) = (1 + (-1)^e)/2 if Kronecker(-19, p) = -1. - Michael Somos, Feb 27 2007
a(n) = 2 * A035171(n) unless n = 0. - Jianing Song, Sep 06 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(19) = 1.441461... . - Amiram Eldar, Dec 16 2023

cp's OEIS Frontend

A279619 Expansion of g.f. of A002652 in powers of the g.f. of A279618.

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A028951 Numbers represented by quadratic form with Gram matrix [ 2, 1; 1, 4 ] (or the Kleinian 2-d lattice, see A002652).

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A205975 a(n) = Fibonacci(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

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A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

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A033715 Number of integer solutions (x, y) to the equation x^2 + 2y^2 = n.

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A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -7.

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A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

A028609 Expansion of (theta_3(z)theta_3(11z) + theta_2(z)theta_2(11z)).