cp's OEIS Frontend

A193009 Decimal expansion of the constant term of the reduction of 2^(-x) by x^2->x+1.

%I A193009 #11 Jan 19 2022 05:38:18
%S A193009 1,2,0,0,6,2,2,0,0,3,9,6,9,0,5,8,5,2,1,0,0,7,3,9,7,6,5,9,0,4,3,8,7,8,
%T A193009 2,6,7,1,7,8,0,5,5,0,0,2,6,3,9,4,8,2,1,8,6,6,5,1,2,1,9,2,1,0,4,2,7,7,
%U A193009 3,8,0,7,9,4,3,8,0,8,3,4,8,6,2,5,3,9,2,5,3,7,1,6,6,4,3,8,8,3,5,1
%N A193009 Decimal expansion of the constant term of the reduction of 2^(-x) by x^2->x+1.
%C A193009 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193009 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193009 Equals 1 + Sum_{k>=1} (-log(2))^k*Fibonacci(k-1)/k!.
%F A193009 Equals (4^(phi-1)*(5+3*sqrt(5)) + sqrt(5))/(5*phi*2^phi), where phi is the golden ratio (A001622). (End)
%e A193009 1.20062200396905852100739765904387826717805500...
%t A193009 f[x_] := 2^(-x); r[n_] := Fibonacci[n];
%t A193009 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193009 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193009 RealDigits[u0, 10]
%Y A193009 Cf. A000045, A001622, A193010, A192232, A193035.
%K A193009 nonn,cons
%O A193009 1,2
%A A193009 _Clark Kimberling_, Jul 14 2011