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A263295 Denominators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...). 3

%I #33 Jul 25 2021 07:02:00

%S 3,5,33,111,933,4913,50585,364717,4460647,40920133,580574377,

%T 6472209467,104581586665,1373040648769,24902871413201,376386726269561,

%U 7573128424949291,129519388933667493,2863373356440803473,54670305859684290279,1317404009250178503245

%N Denominators of the n-th iteration of the alternating continued fraction formed from the positive integers, starting with (1 - ...).

%C As n->inf, A262957(n)/a(n) converges to 0.57663338973018...; this number has a surprisingly elegant standard continued fraction representation: [0; 1, 1, 2, 1, 3, 4, 1, 5, 6, 1, 7, 8, ...].

%H Jinyuan Wang, <a href="/A263295/b263295.txt">Table of n, a(n) for n = 1..448</a>

%e (1-1/(2+1)) = 2/3, so a(1) = 3;

%e (1-1/(2+1/(3-1))) = 3/5, so a(2) = 5;

%e (1-1/(2+1/(3-1/(4+1)))) = 19/33, so a(3) = 33;

%e (1-1/(2+1/(3-1/(4+1/(5-1))))) = 64/111, so a(4) = 111.

%o (PARI) a(n) = if(n%2==0, s=-1, s=1); t=1; while(n>0, t=n+1+s/t; n--; s=-s); denominator(t=1/t)

%o vector(30, n, a(n)) \\ corrected by _Mohammed Sabba_, Dec 22 2015

%Y Same principle as A244279 and A244280 - except here we begin with subtraction rather than addition.

%Y Cf. A262957 (numerators).

%K nonn,frac

%O 1,1

%A _Mohamed Sabba_, Nov 20 2015

%E More terms from _Mohamed Sabba_, Dec 22 2015

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)