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%I #27 May 22 2021 21:28:37
%S 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,
%T 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,
%U 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
%N Decimal expansion of sum(k>=0, 1/2^floor(k*phi) ) where phi = (1+sqrt(5))/2.
%C Number whose digits are obtained from the substitution system (1->(1,0),0->(1)).
%C The n-th term of the continued fraction is 2^Fibonacci(n-2) ) (cf. A000301).
%C This number is known to be transcendental.
%D S. Wolfram, "A new kind of science", p. 913
%H J. L. Davison, <a href="http://www.jstor.org/stable/2041058">A series and its associated continued fraction</a>, Proc. Amer. Math. Soc. 63 (1977), pp. 29-32.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals 1 + A014565.
%e 1.70980344286129131464178739944457559701250220576786...
%t Take[ RealDigits[ Sum[N[1/2^Floor[k*GoldenRatio], 120], {k, 0, 300}]][[1]], 105] (* _Jean-François Alcover_, Jul 28 2011 *)
%o (PARI) phi=(1+sqrt(5))/2; suminf(n=0,2.^-(n*phi\1)) \\ _Charles R Greathouse IV_, Jul 22 2013
%o (PARI) phi=(1+sqrt(5))/2; suminf(n=1, (phi*n\1)/2^n) - 1 /* _Michael Somos_, May 22 2021 */
%K cons,nonn
%O 1,2
%A _Benoit Cloitre_, Aug 19 2002
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