A118822 Numerators of the convergents of the 2-adic continued fraction of zero given by A118821.

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, 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

Offset

1,1

Links

Table of n, a(n) for n=1..108.

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

Formula

Period 8 sequence: [2,-1,0,-1,-2,1,0,1].

G.f.: -x*(x-1)*(x^2+x+2) / ( 1+x^4 ).

a(n) = sqrt((n+1)^2 mod 8)(-1)^floor((n+2)/4). - Wesley Ivan Hurt, Jan 01 2014

Example

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, ...

Maple

A118822:=n->sqrt((n+1)^2 mod 8)*(-1)^floor((n+2)/4); seq(A118822(n), n=1..100); # Wesley Ivan Hurt, Jan 01 2014

Mathematica

Table[Sqrt[Mod[(n+1)^2, 8](-1)^Floor[(n+2)/4], {n, 100}] (* Wesley Ivan Hurt, Jan 01 2014 *)

Prog

(PARI) {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]}
for(n=0, 80, print1(a(n), ", "))
(PARI) {a(n) = [2, -1, 0, -1, -2, 1, 0, 1][(n-1)%8+1]; } \\ Joerg Arndt, Jan 02 2014

Crossrefs

Cf. A118821 (partial quotients), A118823 (denominators).

Sequence in context: A118825 A007877 A098178 * A230074 A230075 A334947

Adjacent sequences: A118819 A118820 A118821 * A118823 A118824 A118825

Keyword

frac,sign

Author

Paul D. Hanna, May 01 2006

Status

approved