A073116 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).

0, 1, 5, 1, 8, 4, 64, 128, 16384, 1048576, 34359738368, 18014398509481984, 1237940039285380274899124224, 11150372599265311570767859136324180752990208, 27606985387162255149739023449108101809804435888681546220650096895197184

Offset: 1

Author

Benoit Cloitre, Aug 19 2002

Keywords

• base
• cofr
• nonn

Comments

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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..20

Crossrefs

Cf. A000045, A000301, A001622, A073115.

Programs

Mathematica

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 *)

Formula

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.

Extensions

More terms from Amiram Eldar, May 08 2022