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Numerators of s(i) = s(i-1) - (1/i)*sign(s(i-1)) with s(1) = 1.
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%I #23 Nov 19 2017 12:52:01

%S 1,1,1,-1,7,-1,13,-9,199,-53,1937,-373,22871,-2869,4231,-547,134845,

%T -1291,2425919,489967,-10595393,-13913,232472561,9379691,-1023947321,

%U 5712079,-2957435363,-89098463,77729577773,93259013,-2326198533397,-139786038869,385098109121

%N Numerators of s(i) = s(i-1) - (1/i)*sign(s(i-1)) with s(1) = 1.

%C Denominators are given in A203811.

%C Similar to harmonic series, but with signs chosen to minimize the absolute value of the next term.

%H Hugo Pfoertner, <a href="/A203810/b203810.txt">Table of n, a(n) for n = 1..200</a>

%H Hugo Pfoertner, <a href="/A203810/a203810.pdf">Illustration of A203810(i)/A203811(i) for i<=100</a>

%H Hugo Pfoertner, <a href="/A203810/a203810_1.pdf">Illustration of A203810(i)/A203811(i) for even i, i<=500</a>

%e s(1)=1, to minimize abs(s(2)) 1/2 has to be subtracted. s(2)=1-1/2=1/2. Similar for s(3) and s(4): s(3)=s(2)-1/3=1/2-1/3=1/6, s(4)=1/6-1/4=-1/12. Since s(4) is negative s(5)=s(4)+1/5=-1/12+1/5=7/60. The numerators of s(1)...s(5) are 1, 1, 1, -1, 7 and the corresponding denominators are 1, 2, 6, 12, 60.

%Y Cf. A203811 (denominators), A203812 (minima of abs(A203810(i)/A203811(i))).

%Y Cf. A001008, A002805 (harmonic numbers).

%K sign,frac

%O 1,5

%A _Hugo Pfoertner_ and _Rainer Rosenthal_, Jan 06 2012