cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Paul D. Hanna, May 01 2006

Keywords

Examples

			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, ...

Links

Crossrefs

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

Programs

  • PARI
    {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]}

Formula

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

A118832 Denominators of the convergents of the 2-adic continued fraction of zero given by A118830.

Original entry on oeis.org

1, 2, -1, -2, 1, 0, 1, 8, -7, -6, -1, -10, 9, 8, 1, 24, -23, -22, -1, -26, 25, 24, 1, 32, -31, -30, -1, -34, 33, 32, 1, 64, -63, -62, -1, -66, 65, 64, 1, 72, -71, -70, -1, -74, 73, 72, 1, 88, -87, -86, -1, -90, 89, 88, 1, 96, -95, -94, -1, -98, 97, 96, 1, 160, -159, -158, -1, -162, 161, 160, 1, 168, -167, -166, -1, -170, 169, 168
Offset: 1

Views

Author

Paul D. Hanna, May 01 2006

Keywords

Examples

			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, ...

Crossrefs

Cf. A080277; A118830 (partial quotients), A118831 (numerators).

Programs

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

Formula

a(4*n) = (-1)^n*2*A080277(n); a(4*n+1) = -(-1)^n*(2*A080277(n)-1); a(4*n+2) = -(-1)^n*(2*A080277(n)-2); a(4*n-1) = (-1)^n.

A118821 2-adic continued fraction of zero, where a(n) = 2 if n is odd, -A006519(n/2) otherwise.

Original entry on oeis.org

2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -16, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -32, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -8, 2, -1, 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -1, 2, -16
Offset: 1

Views

Author

Paul D. Hanna, May 01 2006

Keywords

Comments

Limit of convergents equals zero; only the 6th convergent is indeterminate. Other 2-adic continued fractions of zero are A118824, A118827, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.

Examples

			For n >= 1, convergents A118822(k)/A118823(k) are:
  at k = 4*n: -1/A080277(n);
  at k = 4*n+1: -2/(2*A080277(n)-1);
  at k = 4*n+2: -1/(A080277(n)-1);
  at k = 4*n-1: 0/(-1)^n.
Convergents begin:
  2/1, -1/-1, 0/-1, -1/1, -2/1, 1/0, 0/1, 1/-4,
  2/-7, -1/3, 0/-1, -1/5, -2/9, 1/-4, 0/1, 1/-12,
  2/-23, -1/11, 0/-1, -1/13, -2/25, 1/-12, 0/1, 1/-16,
  2/-31, -1/15, 0/-1, -1/17, -2/33, 1/-16, 0/1, 1/-32, ...

Links

Crossrefs

Cf. A006519, A080277; convergents: A118822/A118823; variants: A118824, A118827, A118830; A100338.

Programs

  • Mathematica
    Array[-2^(IntegerExponent[#, 2] - 1) /. -1/2 -> 2 &, 96] (* Michael De Vlieger, Nov 02 2018 *)
  • PARI
    a(n)=local(p=+2,q=-1);if(n%2==1,p,q*2^valuation(n/2,2))

A118824 2-adic continued fraction of zero, where a(n) = -2 if n is odd, A006519(n/2) otherwise.

Original entry on oeis.org

-2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 8, -2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 16, -2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 8, -2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 32, -2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 8, -2, 1, -2, 2, -2, 1, -2, 4, -2, 1, -2, 2, -2, 1, -2, 16, -2, 1, -2, 2, -2, 1
Offset: 1

Views

Author

Paul D. Hanna, May 01 2006

Keywords

Comments

Limit of convergents equals zero; only the 6th convergent is indeterminate. Other 2-adic continued fractions of zero are: A118821, A118827, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.

Examples

			For n >= 1, convergents A118825(k)/A118826(k):
  at k = 4*n: 1/A080277(n);
  at k = 4*n+1: 2/(2*A080277(n)-1);
  at k = 4*n+2: 1/(A080277(n)-1);
  at k = 4*n-1: 0.
Convergents begin:
  -2/1, -1/1, 0/-1, -1/-1, 2/1, 1/0, 0/1, 1/4,
  -2/-7, -1/-3, 0/-1, -1/-5, 2/9, 1/4, 0/1, 1/12,
  -2/-23, -1/-11, 0/-1, -1/-13, 2/25, 1/12, 0/1, 1/16,
  -2/-31, -1/-15, 0/-1, -1/-17, 2/33, 1/16, 0/1, 1/32, ...

Links

Crossrefs

Cf. A006519, A080277; convergents: A118825/A118826; variants: A118821, A118827, A118830; A100338.

Programs

  • Mathematica
    Array[If[OddQ@ #, -2, 2^(IntegerExponent[#, 2] - 1)] &, 102] (* Michael De Vlieger, Nov 06 2018 *)
  • PARI
    a(n)=local(p=-2,q=+1);if(n%2==1,p,q*2^valuation(n/2,2))

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

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

Views

Author

Paul D. Hanna, May 01 2006

Keywords

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

Examples

			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, ...

Links

Crossrefs

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

Programs

  • Mathematica
    Array[If[OddQ@ #, 1, -2*2^(IntegerExponent[#, 2] - 1)] &, 99] (* Michael De Vlieger, Nov 06 2018 *)
  • PARI
    a(n)=local(p=+1,q=-2);if(n%2==1,p,q*2^valuation(n/2,2))

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)
Showing 1-5 of 5 results.

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