A109169 - OEIS

%I #11 Dec 31 2024 15:39:51

%S 1,4,0,8,4,9,4,2,7,9,2,2,8,9,0,6,9,8,5,7,4,8,4,7,4,2,7,9,0,8,0,6,9,7,

%T 9,9,1,6,1,3,9,9,8,9,5,5,7,8,2,0,5,1,2,8,1,4,6,6,2,6,3,8,1,7,5,2,4,8,

%U 6,2,9,7,7,8,9,9,0,3,0,8,5,3,3,0,1,2,5,6,2,8,5,4,3,0,4,8,6,9,1,8,6,4,8,1,2

%N Decimal expansion of constant x such that the continued fraction expansion of 2*x (A109170) yields the continued fraction expansion of x (A109168) interleaved with positive even numbers.

%e x=1.408494279228906985748474279080697991613998955782051281466263817524862977...

%e The continued fraction expansion of x = A109168:

%e [1; 2, 2, 4, 3, 4, 4, 8, 5, 6, 6, 8, 7, 8, 8, 16, ...];

%e the continued fraction expansion of 2*x = A109170:

%e [2;1, 4,2, 6,2, 8,4, 10,3, 12,4, 14,4, 16,8, 18,5, ...]

%e which equals the continued fraction of x interleaved with even numbers.

%o (PARI) {PQ(n)=if(n%2==1,(n+1)/2,2*PQ(n/2))}

%o {CFM=contfracpnqn(vector(500,n,PQ(n))); CFM[1,1]/CFM[2,1]*1.0}

%Y Cf. A109168 (continued fraction of x), A109170 (continued fraction of 2*x), A109171 (digits of 2*x).

%K nonn,cons

%O 1,2

%A _Paul D. Hanna_, Jun 21 2005