A244279 Numerators of the n-th iteration of the alternating continued fraction of the positive integers, initiated with (1 + ...).

1, 1, 7, 17, 127, 547, 5111, 31865, 358781, 2938437, 38808271, 394282041, 5982064475, 72608885159, 1245025688399, 17581129642961, 336297031232409, 5417081623572649, 114375064174857015, 2069902867431592833, 47819312187294567447, 960634689914268797707

Offset: 1

Author

Mohamed Sabba, Jun 24 2014

Keywords

• nonn
• frac

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...].

Examples

			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.
		

Links

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

Crossrefs

Cf. A244280 (Denominators).

Programs

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

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

Formula

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.

Extensions

More terms from Colin Barker, Jul 20 2014