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A244279
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Numerators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).
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4
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1, 1, 7, 17, 127, 547, 5111, 31865, 358781, 2938437, 38808271, 394282041, 5982064475, 72608885159, 1245025688399, 17581129642961, 336297031232409, 5417081623572649, 114375064174857015, 2069902867431592833, 47819312187294567447, 960634689914268797707
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OFFSET
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1,3
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COMMENTS
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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...].
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..449
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FORMULA
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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.
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EXAMPLE
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a(1) = 1/(1+1) = 1/2;
a(2) = 1/(1+1/(2-1)) = 1/2;
a(3) = 1/(1+1/(2-1/(3+1))) = 7/11;
a(4) = 1/(1+1/(2-1/(3+1/(4-1)))) = 17/27.
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MAPLE
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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
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PROG
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(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)
vector(30, n, a(n)) \\ Colin Barker, Jul 20 2014
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CROSSREFS
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Cf. A244280 (Denominators).
Sequence in context: A266382 A118108 A227506 * A325584 A214149 A147643
Adjacent sequences: A244276 A244277 A244278 * A244280 A244281 A244282
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KEYWORD
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nonn,frac
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AUTHOR
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Mohamed Sabba, Jun 24 2014
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EXTENSIONS
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More terms from Colin Barker, Jul 20 2014
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STATUS
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approved
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