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
Examples
G.f. = q + q^2 - q^3 + q^4 - q^6 + q^8 + q^9 - q^12 + q^16 + q^18 + ...
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.
Programs
-
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
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.
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)
Extensions
More terms from Antti Karttunen, Dec 07 2017