1,7

G.f.: Sum_{k>0} (x^k-2x^(2k)+2x^(4k)-x^(5k))/(1-x^(6k)) = x Product_{k>0} ((1-x^k)(1-x^(6k))^6)/((1-x^(2k))^2(1-x^(3k))^3).

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*u3 +u1*u6 +u2*u3 +4*u2*u6 -u2^2 -3*u3^2 -6*u3*u6 -3*u6^2 - Michael Somos Apr 10 2005

Multiplicative with a(p^e) = (-1)^e if p=2; a(p^e) = 1 if p=3; a(p^e) = 1+e if p == 1 (mod 6); a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).

a(2n)=-a(n). a(3n)=a(n). a(6n+5)=a(12n+11)=0.

Moebius transform is period 6 sequence [ 1, -2, 0, 2, -1, 0, ...]. - Michael Somos Jul 16 2006

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

Expansion of (a(q) - a(q^2)) / 6 = c(q^2)^2 / (3 * c(q)) in powers of q where a(), c() are cubic AGM functions. - Michael Somos Sep 06 2007

Euler transform of period 6 sequence [ -1, 1, 2, 1, -1, -2, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 12^(-1/2) (t/i) g(t) where q = exp(2 pi i t) and g() is g.f. for A122859.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = w * (u + v)^2 - v * (v + w) * (v + 4*w).

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u2 * (u2 - u3 - 4*u6) - (u3 + u6) * (u1 - 3*u3 - 3*u6).

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

(PARI) a(n)=if(n<1, 0, polcoeff(sum(k=0, n, x^k*(1-x^k)^2/(1+x^(2*k)+x^(4*k)), x*O(x^n)), n))

(PARI) a(n)= local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)^6/eta(x^2+A)^2/eta(x^3+A)^3, n))

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-12, d)-if(d%2==0, 2*kronecker(-3, d/2)))) /* Michael Somos May 29 2005 */

Cf. A035178(n)=|a(n)|. A033762(n)=a(2n+1). A033687(n)=a(3n+1).

Sequence in context: A107110 A061197 A035178 this_sequence A113447 A137608 A078807

Adjacent sequences: A093826 A093827 A093828 this_sequence A093830 A093831 A093832

sign,mult

Michael Somos, Apr 17 2004