A118832 -id:A118832 - cp's OEIS frontend

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

Paul D. Hanna, May 01 2006

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.

			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, ...

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

Cf. A006519, A080277; convergents: A118831/A118832; variants: A118821, A118824, A118827; A100338.

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

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

A118831 Numerators of the convergents of the 2-adic continued fraction of zero given by A118830.

Original entry on oeis.org

-1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0, -1, 1, 1, 0, 1, -1, -1, 0

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118831(k)/A118832(k) are:
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, ...

Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A118830 (partial quotients), A118832 (denominators).

{a(n)=local(p=-1,q=+2,v=vector(n,i,if(i%2==1,p,q*2^valuation(i/2,2)))); contfracpnqn(v)[1,1]}

Period 8 sequence: [ -1,-1,0,-1,1,1,0,1].
G.f.: -x*(1+x+x^3)/(1+x^4). [corrected by R. J. Mathar, Jul 22 2009]
a(n) = -a(n-4). - R. J. Mathar, Jul 22 2009

cp's OEIS Frontend

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

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