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%I #28 Jan 18 2016 17:09:46
%S 1,1,7,17,127,547,5111,31865,358781,2938437,38808271,394282041,
%T 5982064475,72608885159,1245025688399,17581129642961,336297031232409,
%U 5417081623572649,114375064174857015,2069902867431592833,47819312187294567447,960634689914268797707
%N Numerators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).
%C As n-->inf, a(n)/A244280(n) converges to 0.628736607098954801603428... ; this number has a surprisingly elegant standard continued fraction representation of [0; 1, 1, 1, 2, 3, 1, 4, 5, 1, 6, 7, 1, 8, 9...].
%H Robert Israel, <a href="/A244279/b244279.txt">Table of n, a(n) for n = 1..449</a>
%F This is the result of taking the numerator of a continued fraction with alternating signs a(n) = 1/(1+1/(2-1/(3+1/(4-...1/(n +/- 1))))), where addition follows an odd number and subtraction follows an even number.
%e a(1) = 1/(1+1) = 1/2;
%e a(2) = 1/(1+1/(2-1)) = 1/2;
%e a(3) = 1/(1+1/(2-1/(3+1))) = 7/11;
%e a(4) = 1/(1+1/(2-1/(3+1/(4-1)))) = 17/27.
%p seq(numer(numtheory:-cfrac([0, [1,1], seq([(-1)^j,j],j=2..n),[(-1)^(n+1),1]])), n = 1..40); # _Robert Israel_, Jan 17 2016
%o (PARI) a(n) = if(n%2==0,s=-1,s=1); t=1; while(n>0, t=n+s/t; n--; s=-s); numerator(t=1/t)
%o vector(30, n, a(n)) \\ _Colin Barker_, Jul 20 2014
%Y Cf. A244280 (Denominators).
%K nonn,frac
%O 1,3
%A _Mohamed Sabba_, Jun 24 2014
%E More terms from _Colin Barker_, Jul 20 2014
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