A073115 Decimal expansion of sum(k>=0, 1/2^floor(k*phi) ) where phi = (1+sqrt(5))/2.

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

Offset: 1

Author

Benoit Cloitre, Aug 19 2002

Keywords

• cons
• nonn

Comments

Number whose digits are obtained from the substitution system (1->(1,0),0->(1)).

The n-th term of the continued fraction is 2^Fibonacci(n-2) (cf. A000301).

This number is known to be transcendental.

Examples

			1.70980344286129131464178739944457559701250220576786...
		

References

• S. Wolfram, "A new kind of science", p. 913

Links

• J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.
• Index entries for transcendental numbers

Programs

Mathematica

Take[ RealDigits[ Sum[N[1/2^Floor[k*GoldenRatio], 120], {k, 0, 300}]][[1]], 105] (* Jean-François Alcover, Jul 28 2011 *)

PARI

phi=(1+sqrt(5))/2; suminf(n=0,2.^-(n*phi\1)) \\ Charles R Greathouse IV, Jul 22 2013

PARI

phi=(1+sqrt(5))/2; suminf(n=1, (phi*n\1)/2^n) - 1 /* Michael Somos, May 22 2021 */

Formula

Equals 1 + A014565.