cp's OEIS Frontend

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 1-6 of 6 results.

A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0
Offset: 0

Views

Author

Michael Somos, Sep 22 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077.

Examples

			G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ...

References

  • D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Links

Crossrefs

Cf. A005928, A006077, A113062, A180318, A227454, A227696.
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.)

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))}

Formula

Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.
Expansion of 2*b(q^4) - b(q) = b(q^2)^3 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...].
Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696.
G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.
a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).

A132973 Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 3, -3, 3, 0, 3, -6, 3, -3, 0, 0, 3, -6, 6, 0, 3, 0, 3, -6, 0, -6, 0, 0, 3, -3, 6, -3, 6, 0, 0, -6, 3, 0, 0, 0, 3, -6, 6, -6, 0, 0, 6, -6, 0, 0, 0, 0, 3, -9, 3, 0, 6, 0, 3, 0, 6, -6, 0, 0, 0, -6, 6, -6, 3, 0, 0, -6, 0, 0, 0, 0, 3, -6, 6, -3, 6, 0, 6, -6
Offset: 0

Views

Author

Michael Somos, Sep 07 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).

Examples

			G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 + ...

Links

Crossrefs

Cf. A033687, A097195, A107760, A132973, A132974, A227696.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]/2, {q, 0, n}]; (* Michael Somos, May 26 2013 *)
  • PARI
    {a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A )), n))};

Formula

Expansion of b(q^2)^2 / b(-q) = b(q) * b(q^4) / b(q^2) in powers of q where b() is a cubic AGM theta function.
Expansion of (a(q^2) + 2 * a(q^4) - a(q)) / 2 = (c(q)^2 - 5 * c(q) * c(q^4) + 4 * c(q^4)^2) / (3 * c(q^2)) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 26 2013
Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, -2, 0, -3, -2, ...].
Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A113447.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * (1 + x^(6*k))).
G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: 1 + 3 * ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = (-1)^n * A107760(n). Convolution inverse of A132974.
a(2*n) = A107760(n). a(2*n + 1) = -3 * A033762(n). a(3*n) = A132973(n). a(3*n + 1) = -3 * A227696(n). - Michael Somos, Oct 31 2015
a(6*n + 1) = -3 * A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. - Michael Somos, Oct 31 2015

A134079 Expansion of q^(-2/3) * c(-q)^2 / 9 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, -2, 5, -4, 8, -6, 14, -8, 14, -10, 21, -16, 20, -14, 28, -16, 31, -18, 40, -20, 32, -28, 42, -24, 38, -32, 62, -28, 44, -30, 56, -40, 57, -34, 70, -36, 72, -38, 70, -48, 62, -52, 85, -44, 68, -46, 112, -56, 74, -50, 100, -64, 80, -64, 98, -56, 108, -58, 124
Offset: 0

Views

Author

Michael Somos, Oct 06 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).

Examples

			G.f. = 1 - 2*x + 5*x^2 - 4*x^3 + 8*x^4 - 6*x^5 + 14*x^6 - 8*x^7 + 14*x^8 - ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 4*q^11 + 8*q^14 - 6*q^17 + 14*q^20 - 8*q^23 + ...

Links

Crossrefs

Cf. A033686, A098098, A134078, A227696, A263773.
Cf. A000700, A000122, A004016, A005882, A005928, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3]^3 / QPochhammer[ -x])^2, {x, 0, n}]; (* Michael Somos, Feb 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[ 1, 3 n + 2] / 3]; (* Michael Somos, Feb 19 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x*O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^9 / ( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) )^3 )^2, n))};
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sigma(3*n + 2) / 3)}; /* Michael Somos, Feb 19 2015 */

Formula

Expansion of ( f(x^3)^3 / f(x) )^2 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6)^18 / (eta(q^2) * eta(q^3)* eta(q^12))^6 in powers of q.
Euler transform of period 12 sequence [ -2, 4, 4, 2, -2, -8, -2, 2, 4, 4, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263773.
a(n) = (-1)^n * A033686(n). 18 * a(n) = A134078(3*n + 2).
From Michael Somos, Feb 19 2015: (Start)
a(2*n + 1) = -2 * A098098(n).
Convolution square of A227696. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/54) * n^2. - Amiram Eldar, Nov 23 2023

A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 14 2006

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 - q^2 - q^4 + 2*q^7 - q^8 + 2*q^13 - 2*q^14 - q^16 + 2*q^19 + ...

Links

Crossrefs

Cf. A033687, A097195, A113448, A121361, A123884.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {1, -1, 0}[[Mod[#, 3, 1]]], Mod[#, 6] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 03 2015 *)
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 2, 0, x^(9/2)] EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 4, 0, x^18] / (2^(3/2) x^(5/4) QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 03 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^4 / (eta(x^2 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), n))};
  • PARI
    {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==2, -1, p==3, 0, p%6==1, e+1, !(e%2))))};

Formula

Expansion of (a(x) - a(x^2) - a(x^3) - 2*a(x^4) + a(x^6) + 2*a(x^12)) / 6 in powers of x where a() is a cubic AGM theta function. - Michael Somos, Aug 03 2015
Expansion of psi(-x) * psi(-x^9) * phi(x^9) / f(-x^6) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 03 2015
Expansion of eta(q) * eta(q^4) * eta(q^18)^4 / (eta(q^2) * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 1, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = a(6*n + 5) = 0.
a(2*n) = -A113448(n). a(6*n + 2) = -A033687(n).
a(3*n + 1) = A227696(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). - Michael Somos, Aug 03 2015

A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jan 29 2008

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 - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...

Links

Crossrefs

Cf. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.
Cf. A035178, A093829, A113447 are same up to sign.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)
  • PARI
    {a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-12, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A))) / 3, n))}; /* Michael Somos, May 06 2015 */

Formula

Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless 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).
G.f.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = -(-1)^n * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n).

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Aug 04 2015

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. = x + x^2 - 2*x^3 - x^4 - 2*x^6 + 2*x^7 + x^8 - 2*x^9 + 2*x^12 + ...

Links

Crossrefs

Cf. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.
Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 1, 0, -1, -1][d%6 + 1] * [0, 1, 0, -2, 0, 1][n\d%6 + 1]))};
  • PARI
    {a(n) = if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, -2, p%6==5, 1-e%2, e+1)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Formula

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = -2, if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024
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