A118824 - OEIS

%I #13 Nov 09 2018 21:57:57

%S -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,

%T 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,

%U -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

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

%C 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.

%H Antti Karttunen, <a href="/A118824/b118824.txt">Table of n, a(n) for n = 1..65537</a>

%e For n >= 1, convergents A118825(k)/A118826(k):

%e   at k = 4*n: 1/A080277(n);

%e   at k = 4*n+1: 2/(2*A080277(n)-1);

%e   at k = 4*n+2: 1/(A080277(n)-1);

%e   at k = 4*n-1: 0.

%e Convergents begin:

%e   -2/1, -1/1, 0/-1, -1/-1, 2/1, 1/0, 0/1, 1/4,

%e   -2/-7, -1/-3, 0/-1, -1/-5, 2/9, 1/4, 0/1, 1/12,

%e   -2/-23, -1/-11, 0/-1, -1/-13, 2/25, 1/12, 0/1, 1/16,

%e   -2/-31, -1/-15, 0/-1, -1/-17, 2/33, 1/16, 0/1, 1/32, ...

%t Array[If[OddQ@ #, -2, 2^(IntegerExponent[#, 2] - 1)] &, 102] (* _Michael De Vlieger_, Nov 06 2018 *)

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

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

%K cofr,sign

%O 1,1

%A _Paul D. Hanna_, May 01 2006