

A244279


Numerators of the nth iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).


4



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

1,3


COMMENTS

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...].


LINKS

Robert Israel, Table of n, a(n) for n = 1..449


FORMULA

This is the result of taking the numerator of a continued fraction with alternating signs a(n) = 1/(1+1/(21/(3+1/(4...1/(n +/ 1))))), where addition follows an odd number and subtraction follows an even number.


EXAMPLE

a(1) = 1/(1+1) = 1/2;
a(2) = 1/(1+1/(21)) = 1/2;
a(3) = 1/(1+1/(21/(3+1))) = 7/11;
a(4) = 1/(1+1/(21/(3+1/(41)))) = 17/27.


MAPLE

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


PROG

(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


CROSSREFS

Cf. A244280 (Denominators).
Sequence in context: A266382 A118108 A227506 * A325584 A214149 A147643
Adjacent sequences: A244276 A244277 A244278 * A244280 A244281 A244282


KEYWORD

nonn,frac


AUTHOR

Mohamed Sabba, Jun 24 2014


EXTENSIONS

More terms from Colin Barker, Jul 20 2014


STATUS

approved



