A028951 -id:A028951 - cp's OEIS frontend

A002652 Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

1, 2, 4, 0, 6, 0, 0, 2, 8, 2, 0, 4, 0, 0, 4, 0, 10, 0, 4, 0, 0, 0, 8, 4, 0, 2, 0, 0, 6, 4, 0, 0, 12, 0, 0, 0, 6, 4, 0, 0, 0, 0, 0, 4, 12, 0, 8, 0, 0, 2, 4, 0, 0, 4, 0, 0, 8, 0, 8, 0, 0, 0, 0, 2, 14, 0, 0, 4, 0, 0, 0, 4, 8, 0, 8, 0, 0, 4, 0, 4, 0, 2, 0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 12, 0, 0, 0, 0, 0, 4, 4, 6, 0

Offset: 0

N. J. A. Sloane

In other words, theta series of lattice with Gram matrix [2, 1; 1, 4].
The number of integer solutions (x, y) to x^2 + x*y + 2*y^2 = n.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 2*x + 4*x^2 + 6*x^4 + 2*x^7 + 8*x^8 + 2*x^9 + 4*x^11 + 4*x^14 + ...
Theta series of lattice with Gram matrix [2, 1; 1, 4] = 1 + 2*q^2 + 4*q^4 + 6*q^8 + 2*q^14 + 8*q^16 + 2*q^18 + 4*q^22 + 4*q^28 + 10*q^32 + 4*q^36 + 8*q^44 + 4*q^46 + 2*q^50 + 6*q^56 + 4*q^58 + 12*q^64 + 6*q^72 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 467, Entry 5(i).

John Cannon, Table of n, a(n) for n = 0..5000
J. H. Conway and N. J. A. Sloane, Complex and integral laminated lattices, Trans. Amer. Math. Soc., 280 (1983), 463-490.
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. A035182, A133827, A028951.
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), this sequence (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43).

A := Basis( ModularForms( Gamma1(14), 1), 85); A[1] + 2*A[2] + 4*A[3] + 6*A[5]; /* Michael Somos, Jun 10 2015 */

f[d_] := KroneckerSymbol[-7, d]; a[n_] := 2*Total[f /@ Divisors[n]]; a[0]=1; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Nov 08 2011, after Michael Somos *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(t2, t3); if( n<1, n==0, t2 = 2 * sum( n=1, (sqrtint( max(0, 4*n - 7)) + 1)\2, x^(n*n - n)); t3 = 1 + 2 * sum( n=1, sqrtint(n), x^(n*n)); polcoeff( t3 * subst(t3, x, x^7) + x^2 * t2 * subst(t2, x, x^7), n))};

{a(n) = my(t); if( n<1, n==0, 2 * issquare(n) + 2 * sum( y=1, sqrtint(n*4\7), 2 * issquare(t = 4*n - 7*y^2) - (t==0)))}; /* Michael Somos, Sep 20 2004 */

{a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1 * A2), n))}; /* Michael Somos, May 28 2005 */

{a(n) = if( n<1, n==0, 2 * qfrep( [ 2, 1; 1, 4], n, 1)[n])}; /* Michael Somos, Jun 03 2005 */

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -7, d)))}; /* Michael Somos, Oct 07 2005 */

G.f.: theta_3(q) * theta_3(q^7) + theta_2(q) * theta_2(q^7).
G.f.: 1 + 2 * Sum_{k>0} Kronecker(-7, k) * x^k / (1 - x^k). - Michael Somos, Mar 17 2012
Expansion of phi(x) * phi(x^7) + 4 * x^2 * psi(x^2) * psi(x^14) = phi(-x) * phi(-x^7) + 4 * x * psi(x) * psi(x^7) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Mar 17 2012
Expansion of ((eta(q) * eta(q^7))^3 + 4 * (eta(q^2) * eta(q^14))^3) / (eta(q) * eta(q^2) * eta(q^7) * eta(q^14)) in powers of q. - Michael Somos, May 28 2005
Moebius transform is period 7 sequence [ 2, 2, -2, 2, -2, -2, 0, ...]. - Michael Somos, Oct 07 2005
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 + 2 * u*w - 4 * u*v - 8 * v*w. - Michael Somos, Sep 20 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) = u1^3*u6 + 2*u2^3*u3 + 18*u1*u3*u6^2 + 18*u2*u3^2*u6 + 6*u1*u2^2*u6 + 3*u1^2*u2*u3 - 3*u2*u3^3 - 18*u2*u3*u6^2 - 6*u1*u6^3 - 9*u1*u3^2*u6 - 6*u1*u2^2*u3 - 6*u1^2*u2*u6. - Michael Somos, Jun 03 2005
From Michael Somos, Mar 17 2012: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (7 t)) = 7^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = 2 * A035182(n) unless n = 0. a(7*n + 5) = a(7*n + 6) = a(9*n + 3) = a(9*n + 6) = 0. a(2*n + 1) = 2 * A133827(n). a(9*n) = a(n). (End)
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(7^e) = 1, b(p^e) = e + 1 if p == 1, 2, 4 (mod 7), b(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, Jun 10 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(7) = 2.3748208... . - Amiram Eldar, Dec 16 2023

A045373 Primes congruent to {0, 1, 2, 4} mod 7.

Original entry on oeis.org

2, 7, 11, 23, 29, 37, 43, 53, 67, 71, 79, 107, 109, 113, 127, 137, 149, 151, 163, 179, 191, 193, 197, 211, 233, 239, 263, 277, 281, 317, 331, 337, 347, 359, 373, 379, 389, 401, 421, 431, 443, 449, 457, 463, 487

Offset: 1

N. J. A. Sloane

nonn

Primes of the form x^2 + xy + 2y^2, discriminant -7. - N. J. A. Sloane, Jun 01 2014
Primes of the form x^2 - xy + 2y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Also, primes which are squares (mod 7) (or, (mod 14): see A191017 for a sequence formerly defined as such). - M. F. Hasler, Jan 15 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Primes in A028951.

Cf. A191017, A003625 (complement).

[p: p in PrimesUpTo(740)|p mod 7 in [0, 1, 2, 4]]; // Vincenzo Librandi, Jul 13 2012

Select[Prime[Range[500]],MemberQ[{0,1,2,4},Mod[#,7]]&] (* Vincenzo Librandi, Jul 13 2012 *)

select(p->issquare(Mod(p,7))&&isprime(p),[1..1000]) \\ M. F. Hasler, Jan 15 2016

A034035 Not of form (x^2+xy+2y^2 with x >= 0, y >= 0).

Original entry on oeis.org

3, 5, 6, 7, 10, 12, 13, 15, 17, 19, 20, 21, 24, 26, 27, 29, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 45, 47, 48, 51, 52, 54, 55, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 73, 75, 76, 78, 80, 82, 83, 84, 85, 87, 89, 90, 91, 93, 94, 95, 96, 97, 101, 102, 103

Offset: 1

N. J. A. Sloane

nonn

Cf. A034035-A034042, A028951, A257346.

Definition corrected by N. J. A. Sloane, Apr 30 2015

A257346 Numbers not of the form x^2+xy+2y^2.

Original entry on oeis.org

3, 5, 6, 10, 12, 13, 15, 17, 19, 20, 21, 24, 26, 27, 30, 31, 33, 34, 35, 38, 39, 40, 41, 42, 45, 47, 48, 51, 52, 54, 55, 57, 59, 60, 61, 62, 65, 66, 68, 69, 70, 73, 75, 76, 78, 80, 82, 83, 84, 85, 87, 89, 90, 91, 93, 94, 95, 96, 97, 101

Offset: 1

N. J. A. Sloane, Apr 30 2015

nonn

Complement of A028951.

A034036 Numbers of the form (x^2+xy+2y^2 with x>=0, y>=0).

Original entry on oeis.org

0, 1, 2, 4, 8, 9, 11, 14, 16, 18, 22, 23, 25, 28, 32, 36, 37, 43, 44, 46, 49, 50, 53, 56, 58, 64, 71, 72, 74, 77, 79, 81, 86, 88, 92, 98, 99, 100, 106, 107, 109, 112, 116, 121, 126, 127, 128, 134, 137, 142, 144, 148, 149, 151, 154, 158, 161, 162, 163, 169

Offset: 1

N. J. A. Sloane

nonn

Cf. A034035-A034042, A028951.

Definition corrected by N. J. A. Sloane, Apr 30 2015

A035248 Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -7 (A035182).

Original entry on oeis.org

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, 163, 169, 172, 175

Offset: 1

N. J. A. Sloane

nonn

Is this A028951 with the initial 0 dropped? [Joerg Arndt, Dec 08 2013]

Cf. A028951. [From R. J. Mathar, Oct 20 2008]

Cf. A035182 (Expansion for m= -7).

Reap[For[n = 1, 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 *)

m=-7; v=direuler(p=2, 301, 1/(1-(kronecker(m, p)*(X-X^2))-X));
for(j=1,#v,if(v[j],print1(j,", ")));

Added more terms, Joerg Arndt, Dec 08 2013

Definition corrected by Georg Fischer, Sep 03 2020

cp's OEIS Frontend

A002652 Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

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A045373 Primes congruent to {0, 1, 2, 4} mod 7.

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A034035 Not of form (x^2+xy+2y^2 with x >= 0, y >= 0).

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A257346 Numbers not of the form x^2+xy+2y^2.

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A034036 Numbers of the form (x^2+xy+2y^2 with x>=0, y>=0).

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Extensions

A035248 Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m= -7 (A035182).

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Extensions

A035248 Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m= -7 (A035182).