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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 2, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
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).

Examples

			G.f. = 1 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

References

  • Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Links

Crossrefs

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */
  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */
  • PARI
    {a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */
  • PARI
    {a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */

Formula

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
From Michael Somos, Sep 18 2004: (Start)
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).
6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009

A129576 Expansion of phi(x) * chi(x) * psi(-x^3) in powers of x where phi(), chi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 0, 2, 3, 2, 0, 1, 6, 2, 0, 2, 0, 2, 0, 3, 6, 0, 0, 2, 3, 2, 0, 2, 6, 2, 0, 0, 0, 4, 0, 2, 3, 2, 0, 2, 6, 0, 0, 1, 6, 2, 0, 4, 0, 2, 0, 0, 6, 2, 0, 2, 0, 2, 0, 3, 6, 2, 0, 2, 0, 0, 0, 2, 9, 2, 0, 0, 6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, 6, 4, 0, 0, 3, 4
Offset: 0

Views

Author

Michael Somos, Apr 23 2007

Keywords

Comments

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). - Michael Somos, Jun 28 2017

Examples

			G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 3*x^5 + 2*x^6 + x^8 + 6*x^9 + 2*x^10 + ...
G.f. = q + 3*q^4 + 2*q^7 + 2*q^13 + 3*q^16 + 2*q^19 + q^25 + 6*q^28 + ...

Links

Crossrefs

Cf. A033716, A093602, A096936, A112298, A122861.
Cf. A000122, A000700, A004016, A005882, A005928, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := Length @ FindInstance[ x^2 + 3 y^2 == 3 n + 1, {x, y}, Integers, 10^9] / 2; (* Michael Somos, Sep 03 2016 *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# == 3, Boole[#2 == 0], # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 2, (1 + (-1)^#2)/2, True, #2 + 1] & @@@ FactorInteger[3 n + 1])]; (* Michael Somos, Jun 28 2017 *)
  • PARI
    {a(n) = if( n<0, 0, n = 3*n + 1; sumdiv(n, d, kronecker(-3, d) * [0, 1, -2, 1] [n/d%4 + 1] ))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 3*(1-e%2), p==3, 0, p%3==2, 1-e%2, e+1)))}; /* Michael Somos, Jun 28 2017 */

Formula

From Michael Somos, Jun 28 2017: (Start)
Expansion of q^(-1/3) * (2*c(q) + c(-q)) / 3 = q^(-1/3) * (c(q) + 2*c(q^4)) / 3 in powers of q where c() is a cubic AGM theta function.
Expansion of (a(q) - a(q^3) + 2*a(q^4) - 2*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q^(-1/3) * eta(q^2)^7 * eta(q^3) * eta(q^12) / (eta(q)^3 * eta(q^4)^3 * eta(q^6)) in powers of q. (End)
Euler transform of period 12 sequence [3, -4, 2, -1, 3, -4, 3, -1, 2, -4, 3, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative and b(2^e) = 3 * (1 + (-1)^e) / 2 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1 + (-1)^e)/2 if p == 2 (mod 3).
a(n) = A096936(3*n + 1) = A112298(3*n + 1).
2 * a(n) = A033716(3*n + 1). - Michael Somos, Sep 03 2016
a(n) = (-1)^n * A122161(n). - Michael Somos, Jun 28 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(3) = 1.813799... (A093602). - Amiram Eldar, Dec 29 2023

A244375 Expansion of (a(q) + 3*a(q^2) - 4*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 1, -3, 0, 3, 2, 3, 1, 0, 0, -3, 2, 6, 0, -3, 0, 3, 2, 0, 2, 0, 0, 3, 1, 6, 1, -6, 0, 0, 2, 3, 0, 0, 0, -3, 2, 6, 2, 0, 0, 6, 2, 0, 0, 0, 0, -3, 3, 3, 0, -6, 0, 3, 0, 6, 2, 0, 0, 0, 2, 6, 2, -3, 0, 0, 2, 0, 0, 0, 0, 3, 2, 6, 1, -6, 0, 6, 2, 0, 1, 0, 0
Offset: 1

Views

Author

Michael Somos, Jun 26 2014

Keywords

Comments

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).

Examples

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

Links

Crossrefs

Cf. A033687, A033762, A093602, A093829, A112298, A122861, A244339.
Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 1), 82); A[2] + 3*A[3] + A[4] - 3*A[5]; /* Michael Somos, Jan 17 2015 */
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ Mod[ n/d, 2] {1, 3, 0, -3, -1, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^8 QPochhammer[ q^3] QPochhammer[ q^12]^4 / (QPochhammer[ q]^3 QPochhammer[ q^4]^4 QPochhammer[ q^6]^4), {q, 0, n}];
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (2^(1/2) q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, (n/d%2) * [0, 1, 3, 0, -3, -1][d%6 + 1]))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x + A)^3 * eta(x^4 + A)^4 * eta(x^6 + A)^4), n))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [0, 1, 4, 1][k%4 + 1], x * O(x^n)), n))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 - x^(2*k)) * [0, 1, 3, 0, -3, -1][k%6 + 1], x * O(x^n)), n))};
  • PARI
    {a(n) = my(A);  if( n<1, 0, A = factor(n); prod( j=1, matsize(A)[1], if( p = A[j,1], e = A[j,2]; if( p==2, 3 * (-1)^(e+1), if( p==3, 1, if( p%6 == 1, e+1, (1 + (-1)^e) / 2))))))};

Formula

Expansion of (b(q) - b(q^4)) * (b(q) - 2*b(q^4)) / (3* b(q^2)) = b(q^2)^2 * (b(q^4) - b(q)) / (3 * b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of q * phi(q)^2 * psi(q^6)^2 / (psi(q) * psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * chi(q)^3 * phi(-q^2) * psi(-q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jan 17 2015
Expansion of q * f(-q) * f(q, q^5)^4 / f(-q^3)^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Jan 17 2015
Expansion of eta(q^2)^8 * eta(q^3) * eta(q^12)^4 / (eta(q)^3 * eta(q^4)^4 * eta(q^6)^4) in powers of q. - Michael Somos, Jan 17 2015
a(n) is multiplicative with a(2^e) = 3 * (-1)^(e+1) 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).
Euler transform of period 12 sequence [ 3, -5, 2, -1, 3, -2, 3, -1, 2, -5, 3, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, -6, -1, 0, 1, 6, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244339.
a(2*n) = 3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0.
a(n) = -(-1)^n * A112298(n). - Michael Somos, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024
Showing 1-3 of 3 results.

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