A118827 2-adic continued fraction of zero, where a(n) = 1 if n is odd, otherwise -2*A006519(n/2).

1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -32, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -64, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -32, 1, -2, 1

Offset

1,2

Comments

Limit of convergents equals zero; only the 6th convergent is indeterminate. Other 2-adic continued fractions of zero are: A118821, A118824, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.

Multiplicative because both A006519 and A165326 are. - Andrew Howroyd, Aug 01 2018

Links

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

Formula

a(n) = A165326(n) * A006519(n). - Andrew Howroyd, Aug 01 2018

From Amiram Eldar, Oct 28 2023: (Start)

Multiplicative with a(2^e) = -2^e, and a(p^e) = 1 for an odd prime p.

Dirichlet g.f.: zeta(s) * (1 - 2^(1-s) + 1/(2-2^s)).

Sum_{k=1..n} a(k) ~ (-1/(2*log(2))) * n *(log(n) + gamma - log(2)/2 - 1), where gamma is Euler's constant (A001620). (End)

Example

For n >= 1, convergents A118828(k)/A118829(k):
  at k = 4*n: -1/(2*A080277(n));
  at k = 4*n+1: -1/(2*A080277(n)-1);
  at k = 4*n+2: -1/(2*A080277(n)-2);
  at k = 4*n-1: 0.
Convergents begin:
  1/1, -1/-2, 0/-1, -1/2, -1/1, 1/0, 0/1, 1/-8,
  1/-7, -1/6, 0/-1, -1/10, -1/9, 1/-8, 0/1, 1/-24,
  1/-23, -1/22, 0/-1, -1/26, -1/25, 1/-24, 0/1, 1/-32,
  1/-31, -1/30, 0/-1, -1/34, -1/33, 1/-32, 0/1, 1/-64, ...

Mathematica

Array[If[OddQ@ #, 1, -2*2^(IntegerExponent[#, 2] - 1)] &, 99] (* Michael De Vlieger, Nov 06 2018 *)

Prog

(PARI) a(n)=local(p=+1, q=-2); if(n%2==1, p, q*2^valuation(n/2, 2))

Crossrefs

Cf. A001620, A006519, A080277; convergents: A118828/A118829; variants: A118821, A118824, A118830; A100338, A165326.

Sequence in context: A335852 A353751 A003484 * A118830 A055975 A006519

Adjacent sequences: A118824 A118825 A118826 * A118828 A118829 A118830

Keyword

cofr,sign,mult

Author

Paul D. Hanna, May 01 2006

Status

approved