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

%I A118821 #13 Nov 03 2018 18:48:47
%S A118821 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 A118821 -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 A118821 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
%N A118821 2-adic continued fraction of zero, where a(n) = 2 if n is odd, -A006519(n/2) otherwise.
%C A118821 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.
%H A118821 Antti Karttunen, <a href="/A118821/b118821.txt">Table of n, a(n) for n = 1..65537</a>
%e A118821 For n >= 1, convergents A118822(k)/A118823(k) are:
%e A118821   at k = 4*n: -1/A080277(n);
%e A118821   at k = 4*n+1: -2/(2*A080277(n)-1);
%e A118821   at k = 4*n+2: -1/(A080277(n)-1);
%e A118821   at k = 4*n-1: 0/(-1)^n.
%e A118821 Convergents begin:
%e A118821   2/1, -1/-1, 0/-1, -1/1, -2/1, 1/0, 0/1, 1/-4,
%e A118821   2/-7, -1/3, 0/-1, -1/5, -2/9, 1/-4, 0/1, 1/-12,
%e A118821   2/-23, -1/11, 0/-1, -1/13, -2/25, 1/-12, 0/1, 1/-16,
%e A118821   2/-31, -1/15, 0/-1, -1/17, -2/33, 1/-16, 0/1, 1/-32, ...
%t A118821 Array[-2^(IntegerExponent[#, 2] - 1) /. -1/2 -> 2 &, 96] (* _Michael De Vlieger_, Nov 02 2018 *)
%o A118821 (PARI) a(n)=local(p=+2,q=-1);if(n%2==1,p,q*2^valuation(n/2,2))
%Y A118821 Cf. A006519, A080277; convergents: A118822/A118823; variants: A118824, A118827, A118830; A100338.
%K A118821 cofr,sign
%O A118821 1,1
%A A118821 _Paul D. Hanna_, May 01 2006