A100338 -id:A100338 - cp's OEIS frontend

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

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, 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

			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, ...

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

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

Array[If[OddQ@ #, 1, -2*2^(IntegerExponent[#, 2] - 1)] &, 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))

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)

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

Original entry on oeis.org

-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))

A100865 Records in the continued fraction expansion A100864.

Original entry on oeis.org

1, 4, 74, 8457, 186282390, 430917181166219, 41151315877490090952542206046, 13991468824374967392702752173757116934238293984253807017

Offset: 1

Paul D. Hanna, Nov 21 2004

nonn

These terms are doubly exponential. The next term has 106 digits. Positions of these large partial quotients are given by A100866. Hans Havermann has observed that the ratio of number-of-digits to position number is about 1.03, nearly equal to the reciprocal of Lochs constant.

Dzmitry Badziahin and Jeffrey Shallit, An Unusual Continued Fraction, arXiv:1505.00667 [math.NT], 2015.
Dzmitry Badziahin and Jeffrey Shallit, An unusual continued fraction, Proc. Amer. Math. Soc. 144 (2016), 1887-1896.

Cf. A006519, A100338, A100863, A100864, A100866.

A100866 Positions of records in the continued fraction expansion A100864.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 21, 35, 71, 143, 291, 635, 1407, 2979, 6101, 12339, 25019, 50413, 101339, 202793, 405745, 811365, 1624043, 3249293, 6502711, 13011309

Offset: 1

Paul D. Hanna, Nov 21 2004

nonn

Records form A100865 and are doubly exponential. These terms were independently computed by Robert G. Wilson v and Hans Havermann.

Cf. A100338, A100863, A100864, A100865.

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

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

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A100865 Records in the continued fraction expansion A100864.

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A100866 Positions of records in the continued fraction expansion A100864.

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