OFFSET
1,1
COMMENTS
Equals +1 if n is an even power of 2 (2^0, 2^2, 2^4,...), -1 if n is an odd power of 2 (2^1, 2^3, 2^5,..) and zero anywhere else.
Mobius transform of A035263. - R. J. Mathar, Jul 14 2012
LINKS
Mats Granvik (first 220 terms) & Antti Karttunen, Table of n, a(n) for n = 1..65536
FORMULA
a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = 0^e if p>2. - Michael Somos, Jul 05 2009
G.f. A(x) satisfies x = A(x) + A(x^2).
Dirichlet g.f.: (1 + 2^(-s))^(-1). - Michael Somos, Jul 05 2009
a(1) = 1, after which: a(2n) = -a(n), a(2n+1) = 0. - Antti Karttunen, Jul 24 2017
EXAMPLE
x - x^2 + x^4 - x^8 + x^16 - x^32 + x^64 - x^128 + x^256 - x^512 + ...
MAPLE
a:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
seq(a(n), n=1..95); # Alois P. Heinz, Feb 18 2024
MATHEMATICA
nn = 95; a = PadRight[{1, 1}, nn, 0]; Inverse[Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]][[All, 1]] (* Mats Granvik, Jul 24 2017 *)
PROG
(PARI) {a(n) = if( n < 2, n == 1, - a(n / 2))} /* Michael Somos, Jul 05 2009 */
(Scheme) (define (A154269 n) (cond ((= 1 n) 1) ((even? n) (* -1 (A154269 (/ n 2)))) (else 0))) ;; Antti Karttunen, Jul 24 2017
CROSSREFS
KEYWORD
sign,mult
AUTHOR
Mats Granvik, Jan 06 2009
EXTENSIONS
Alternative description added to the name by Antti Karttunen, Jul 24 2017
STATUS
approved