A118830 2-adic continued fraction of zero, where a(n) = -1 if n is odd, 2*A006519(n/2) otherwise.

-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, A118827. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.

Links

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

Example

For n >= 1, convergents A118831(k)/A118832(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^IntegerExponent[#, 2]] &, 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. A006519, A080277; convergents: A118831/A118832; variants: A118821, A118824, A118827; A100338.

Sequence in context: A353751 A003484 A118827 * A055975 A006519 A356166

Adjacent sequences: A118827 A118828 A118829 * A118831 A118832 A118833

Keyword

cofr,sign

Author

Paul D. Hanna, May 01 2006

Status

approved