OFFSET
0,1
COMMENTS
If binary(n) has adjacent 1 bits then a(n) = 0 else a(n) = (-1)^A000120(n).
Fibbinary numbers (A003714) gives the numbers n for which a(n) = A106400(n). - Antti Karttunen, May 30 2017
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..10922
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238, DOI: 10.1007/978-3-642-23283-1_15.
FORMULA
A024490(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 1.
A005252(n) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = -1.
A027935(n-1) = number of solutions to 2^n <= k < 2^(n+1) and a(k) = 0.
G.f. A(x) satisfies A(x) = A(x^2) - x * A(x^4).
G.f. B(x) of A000621 satisfies B(x) = x * A(x^2) / A(x).
EXAMPLE
G.f. = 1 - x - x^2 - x^4 + x^5 - x^8 + x^9 + x^10 - x^16 + x^17 + x^18 + ...
MATHEMATICA
m = 100; A[_] = 1;
Do[A[x_] = A[x^2] - x A[x^4] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2019 *)
PROG
(PARI) {a(n) = if( n<1, n==0, if( n%2, if( n%4 > 1, 0, -a((n-1)/4) ), a(n/2) ) )};
(PARI) {a(n) = my(A, m); if( n<0, 0, m = 1; A = 1 + O(x); while( m<=n, m *= 2; A = subst(A, x, x^2) - x * subst(A, x, x^4) ); polcoeff(A, n)) };
(Scheme) (define (A132971 n) (cond ((zero? n) 1) ((even? n) (A132971 (/ n 2))) ((= 1 (modulo n 4)) (- (A132971 (/ (- n 1) 4)))) (else 0))) ;; Antti Karttunen, May 30 2017
(Python)
from sympy import mobius, prime, log
import math
def A(n): return n - 2**int(math.floor(log(n, 2)))
def b(n): return n + 1 if n<2 else prime(1 + (len(bin(n)[2:]) - bin(n)[2:].count("1"))) * b(A(n))
def a(n): return mobius(b(n)) # Indranil Ghosh, May 30 2017
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 17 2007, Sep 19 2007
STATUS
approved