cp's OEIS Frontend

A193078 Decimal expansion of the coefficient of x in the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622) (negated).

%I A193078 #12 Jan 19 2022 05:37:39
%S A193078 3,9,6,8,2,1,7,6,2,2,5,4,6,3,9,9,6,6,8,6,8,3,1,5,6,0,2,9,7,3,5,3,0,1,
%T A193078 9,7,1,6,7,6,0,2,7,5,4,8,5,1,5,4,4,8,5,3,3,0,5,9,9,0,1,0,9,9,9,9,6,1,
%U A193078 9,7,5,4,0,3,0,0,6,9,5,4,9,7,6,3,0,7,2,8,7,1,9,2,0,9,6,8,0,7,7,7
%N A193078 Decimal expansion of the coefficient of x in the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622) (negated).
%C A193078 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193078 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193078 Equals Sum_{k>=0} (-log(phi))^k*Fibonacci(k)/k!.
%F A193078 Equals -(phi^sqrt(5) - 1)/(sqrt(5)*phi^phi). (End)
%e A193078 -0.39682176225463996686831560297353019716760...
%t A193078 t = GoldenRatio
%t A193078 f[x_] := t^(-x); r[n_] := Fibonacci[n];
%t A193078 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193078 u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t A193078 RealDigits[u1, 10]
%Y A193078 Cf. A000045, A001622, A193010, A192232, A193077.
%K A193078 nonn,cons
%O A193078 0,1
%A A193078 _Clark Kimberling_, Jul 15 2011