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%I #10 Jan 19 2022 05:38:10
%S 5,4,0,6,8,2,6,4,1,9,5,8,4,8,0,3,8,3,7,7,7,4,1,0,5,5,2,7,2,4,2,2,1,3,
%T 0,1,2,4,8,5,3,2,6,9,1,1,1,1,6,8,3,2,4,5,8,9,2,4,2,2,0,4,6,0,0,1,1,2,
%U 4,2,6,6,3,6,2,3,0,3,2,9,8,4,8,6,1,1,9,1,3,0,5,0,8,7,2,7,3,3,7,3
%N Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.
%C Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F From _Amiram Eldar_, Jan 19 2022: (Start)
%F Equals Sum_{k>=0} (-log(2))^k*Fibonacci(k)/k!.
%F Equals -(2^sqrt(5) - 1)/(sqrt(5)*2^phi), where phi is the golden ratio (A001622). (End)
%e -0.540682641958480383777410552724221301248532691111...
%t f[x_] := 2^(-x); r[n_] := Fibonacci[n];
%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t RealDigits[u1, 10]
%Y Cf. A000045, A001622, A193010, A192232, A193009.
%K nonn,cons
%O 0,1
%A _Clark Kimberling_, Jul 14 2011
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