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%I #17 Dec 14 2023 05:25:02
%S 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,
%T -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,
%U -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
%N Numerators of the convergents of the 2-adic continued fraction of zero given by A118827.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).
%F Period 8 sequence: [1, -1, 0, -1, -1, 1, 0, 1].
%F G.f.: (1 - x - x^3)/(1 + x^4).
%F Assuming offset 0 with a(0) = 1, then a has the g.f. (1 + x - x^2)/(1 + x^4) and a(n) = signum(mods(n+1, 4)*mods(n+1, 8)), where mods(a, b) is the symmetric modulo function. - _Peter Luschny_, Oct 13 2020
%e For n>=1, convergents A118828(k)/A118829(k) are:
%e at k = 4*n: -1/(2*A080277(n));
%e at k = 4*n+1: -1/(2*A080277(n)-1);
%e at k = 4*n+2: -1/(2*A080277(n)-2);
%e at k = 4*n-1: 0/(-1)^n.
%e Convergents begin:
%e 1/1, -1/-2, 0/-1, -1/2, -1/1, 1/0, 0/1, 1/-8,
%e 1/-7, -1/6, 0/-1, -1/10, -1/9, 1/-8, 0/1, 1/-24,
%e 1/-23, -1/22, 0/-1, -1/26, -1/25, 1/-24, 0/1, 1/-32,
%e 1/-31, -1/30, 0/-1, -1/34, -1/33, 1/-32, 0/1, 1/-64, ...
%p seq(signum(mods(n+1, 4)*mods(n+1, 8)), n=1..100); # _Peter Luschny_, Oct 13 2020
%o (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]}
%Y Cf. A118827 (partial quotients), A118829 (denominators).
%K frac,sign,easy
%O 1,1
%A _Paul D. Hanna_, May 01 2006
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