A118822 -id:A118822 - cp's OEIS frontend

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

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

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 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.

			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, ...

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

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

Array[-2^(IntegerExponent[#, 2] - 1) /. -1/2 -> 2 &, 96] (* Michael De Vlieger, Nov 02 2018 *)

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

A118825 Numerators of the convergents of the 2-adic continued fraction of zero given by A118824.

Original entry on oeis.org

-2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1, 0, 1, -2, -1, 0, -1, 2, 1

Offset: 1

Paul D. Hanna, May 01 2006

			For n>=1, convergents A118825(k)/A118826(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, ...

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

Cf. A118824 (partial quotients), A118826 (denominators), A118822, A230075 (start with a(5)).

A118825:=n->sqrt((n+1)^2 mod 8))*(-1)^floor((n+3)/4); seq(A118825(n), n=1..100); # Wesley Ivan Hurt, Jan 04 2014

Table[Sqrt[Mod[(n+1)^2, 8]](-1)^Floor[(n+3)/4], {n, 100}] (* Wesley Ivan Hurt, Jan 04 2014 *)
PadRight[{},120,{-2,-1,0,-1,2,1,0,1}] (* Harvey P. Dale, May 26 2020 *)

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

Period 8 sequence: [ -2,-1,0,-1,2,1,0,1].
G.f.: -x*(1+x)*(x^2-x+2) / ( 1+x^4 ).
a(n) = sqrt((n+1)^2 mod 8)*(-1)^floor((n+3)/4). - Wesley Ivan Hurt, Jan 04 2014

A118823 Denominators of the convergents of the 2-adic continued fraction of zero given by A118821.

Original entry on oeis.org

1, -1, -1, 1, 1, 0, 1, -4, -7, 3, -1, 5, 9, -4, 1, -12, -23, 11, -1, 13, 25, -12, 1, -16, -31, 15, -1, 17, 33, -16, 1, -32, -63, 31, -1, 33, 65, -32, 1, -36, -71, 35, -1, 37, 73, -36, 1, -44, -87, 43, -1, 45, 89, -44, 1, -48, -95, 47, -1, 49, 97, -48, 1, -80, -159, 79, -1, 81, 161, -80, 1, -84, -167, 83, -1, 85, 169, -84, 1, -92

Offset: 1

Paul D. Hanna, May 01 2006

			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, ...

Cf. A080277; A118821 (partial quotients), A118822 (numerators).

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

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

cp's OEIS Frontend

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

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A118825 Numerators of the convergents of the 2-adic continued fraction of zero given by A118824.

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A118823 Denominators of the convergents of the 2-adic continued fraction of zero given by A118821.

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