A087810 First differences of A029931.

1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 1, 1, 1, -5, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 1, 1, 1, -9, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 1, 1, 1, -5, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 1, 1, 1, -14, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0, 1, 1, 1, -5, 1, 1, 1, 0, 1, 1, 1, -2, 1, 1, 1, 0

Offset: 1

Author

Ralf Stephan, Oct 16 2003

Keywords

• sign
• easy
• mult

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385

Crossrefs

Cf. A000217, A007814, A029931.

Programs

Mathematica

Differences[ Table[ (bits = IntegerDigits[n, 2]) . Reverse[ Range[ Length[bits]]], {n, 0, 92}]] (* Jean-François Alcover, Sep 03 2012 *)

PARI

a(n)=if(n<1,0,if(n%2==0,if(n%4,1,1-valuation(n,2)*(valuation(n,2)-1)/2),1))

PARI

a(n)=polcoeff(sum(k=0, floor(log(n)/log(2)), (k+1)*x^2^k/(1+x^2^k)) + O(x^(n+1)), n)

Scheme

(define (A087810 n) (- (A029931 n) (A029931 (- n 1))))
(define (A029931 n) (let loop ((n n) (i 1) (s 0)) (cond ((zero? n) s) ((odd? n) (loop (/ (- n 1) 2) (+ 1 i) (+ s i))) (else (loop (/ n 2) (+ 1 i) s)))))
;; Antti Karttunen, Nov 18 2017

Formula

a(4n) = 1 - T(v_2(n)), else a(n) = 1, where T = A000217 (triangular numbers) and v_2 = A007814 (exponent of 2 in factorization of n).

G.f.: Sum_{k>=0} (k+1)t/(1+t), where t = x^2^k.

Multiplicative with a(2^e) = 1 - A000217(e-1), a(p^e) = 1 otherwise. - Mitch Harris, May 17 2005

Dirichlet g.f.: zeta(s) * (1 - 1/(2^s-1)^2). - Amiram Eldar, Oct 31 2023