OFFSET
1,2
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.
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.
FORMULA
Equals 1 + A014565.
EXAMPLE
1.70980344286129131464178739944457559701250220576786...
MATHEMATICA
Take[ RealDigits[ Sum[N[1/2^Floor[k*GoldenRatio], 120], {k, 0, 300}]][[1]], 105] (* Jean-François Alcover, Jul 28 2011 *)
PROG
(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 */
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Benoit Cloitre, Aug 19 2002
STATUS
approved