A244611 Expansion of (phi(q) + phi(q^2) - phi(q^3) - phi(q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, 1, -1, 1, 0, -1, 0, 1, 1, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 1

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Michael Somos, Introduction to Ramanujan theta functions.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Formula

Multiplicative with a(2^e) = 1, a(3^e) = (-1)^e, and a(p^e) = (1 + (-1)^e) / 2 if p>3.

G.f.: (theta_3(q) + theta_3(q^2) - theta_3(q^3) - theta_3(q^6)) / 2.

a(2*n) = a(n). a(2*n + 1) = A214505(n). a(3*n) = -a(n). a(3*n + 1) = A089801(n). a(6*n + 5) = 0.

From Amiram Eldar, Sep 12 2023: (Start)

Dirichlet g.f.: (1 + 1/2^s) * (1 - 1/3^s) * zeta(2*s).

Sum_{k=1..n} a(k) ~ c * sqrt(n), where c = 1 + 1/sqrt(2) - 1/sqrt(3) - 1/sqrt(6) = 0.721508... . (End)

Example

G.f. = q + q^2 - q^3 + q^4 - q^6 + q^8 + q^9 - q^12 + q^16 + q^18 + ...

Prog

(PARI) {a(n) = issquare(n) + issquare(2*n) - issquare(3*n) - issquare(6*n)};
(PARI) {a(n) = if( n<1, 0, n/= 2^valuation(n, 2); issquare(n) - issquare(n*3))};
(PARI) {a(n) = local(A); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, (-1)^e, !(e%2))))))};
(Scheme)
;; Based on the given multiplicative formula, and using the memoization-macro definec:
(definec (A244611 n) (cond ((= 1 n) n) ((even? n) (A244611 (A000265 n))) ((zero? (modulo n 3)) (* (expt -1 (A067029 n)) (A244611 (A028234 n)))) (else (* 1/2 (+ 1 (expt -1 (A067029 n))) (A244611 (A028234 n))))))
;; Antti Karttunen, Dec 07 2017

Crossrefs

Cf. A089801, A214505.

Sequence in context: A174852 A341517 A065333 * A189289 A270885 A353682

Adjacent sequences: A244608 A244609 A244610 * A244612 A244613 A244614

Keyword

sign,easy,mult

Author

Michael Somos, Jul 01 2014

Extensions

More terms from Antti Karttunen, Dec 07 2017

Status

approved