A073116 - OEIS

%I #11 May 08 2022 08:48:32

%S 0,1,5,1,8,4,64,128,16384,1048576,34359738368,18014398509481984,

%T 1237940039285380274899124224,

%U 11150372599265311570767859136324180752990208,27606985387162255149739023449108101809804435888681546220650096895197184

%N Continued fraction expansion of S/2 where S = Sum_{k>=0} 1/2^floor(k*phi) (A073115) and phi is the golden ratio (1+sqrt(5))/2 (A001622).

%C The number S is the number whose digits are obtained from the substitution system (1->(1,0),0->(1)). The n-th term of the continued fraction expansion for S is 2^Fibonacci(n-2) (cf. A000301). This number S is known to be transcendental. The continued fraction of S/2^m follows the same kind of rule as for S/2.

%H Amiram Eldar, <a href="/A073116/b073116.txt">Table of n, a(n) for n = 1..20</a>

%F If n>2, a(2n+1) = 2^(F(2n-1)+1) and a(2n)= 2^(F(2n-2)-1), where F(n) is the n-th Fibonacci number.

%t a[1] = 0; a[2] = 1; a[3] = 5; a[n_] := 2^(Fibonacci[n - 2] - (-1)^n); Array[a, 15] (* _Amiram Eldar_, May 08 2022 *)

%Y Cf. A000045, A000301, A001622, A073115.

%K base,cofr,nonn

%O 1,3

%A _Benoit Cloitre_, Aug 19 2002

%E More terms from _Amiram Eldar_, May 08 2022