cp's OEIS Frontend

A193035 Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.

%I A193035 #14 Aug 04 2024 10:14:15
%S A193035 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 A193035 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 A193035 4,2,6,6,3,6,2,3,0,3,2,9,8,4,8,6,1,1,9,1,3,0,5,0,8,7,2,7,3,3,7,2,6,3
%N A193035 Decimal expansion of the coefficient of x in the reduction of 2^(-x) by x^2->x+1.
%C A193035 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193035 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193035 Equals Sum_{k>=0} (-log(2))^k*Fibonacci(k)/k!.
%F A193035 Equals -(2^sqrt(5) - 1)/(sqrt(5)*2^phi), where phi is the golden ratio (A001622). (End)
%e A193035 -0.540682641958480383777410552724221301248532691111...
%t A193035 f[x_] := 2^(-x); r[n_] := Fibonacci[n];
%t A193035 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193035 u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t A193035 RealDigits[u1, 10]
%Y A193035 Cf. A000045, A001622, A193010, A192232, A193009.
%K A193035 nonn,cons
%O A193035 0,1
%A A193035 _Clark Kimberling_, Jul 14 2011
%E A193035 a(99) corrected by _Georg Fischer_, Aug 04 2024