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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

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

M. 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.

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: A054525 A174852 A065333 * A189289 A270885 A127972

Adjacent sequences:  A244608 A244609 A244610 * A244612 A244613 A244614

KEYWORD

sign,mult

AUTHOR

Michael Somos, Jul 01 2014

EXTENSIONS

More terms from Antti Karttunen, Dec 07 2017

STATUS

approved

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Last modified September 17 04:49 EDT 2019. Contains 327119 sequences. (Running on oeis4.)