cp's OEIS Frontend

A193077 Decimal expansion of the constant term of the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

%I A193077 #11 Jan 19 2022 05:37:58
%S A193077 1,1,0,1,1,1,1,4,8,3,4,8,5,8,7,1,8,3,8,0,2,6,7,2,0,6,1,9,8,4,0,9,9,7,
%T A193077 5,8,1,1,9,0,2,8,5,1,1,9,0,3,3,6,2,5,4,5,1,7,2,5,8,3,9,6,4,1,3,8,0,7,
%U A193077 6,5,2,2,9,5,6,0,0,1,7,8,1,3,5,3,1,8,5,1,7,9,8,7,6,8,4,1,5,9,0,0
%N A193077 Decimal expansion of the constant term of the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).
%C A193077 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193077 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193077 Equals 1 + Sum_{k>=1} (-log(phi))^k*Fibonacci(k-1)/k!.
%F A193077 Equals (1 + phi^(2*phi+1))/(sqrt(5)*phi^(phi+1)). (End)
%e A193077 1.101111483485871838026720619840...
%t A193077 t = GoldenRatio
%t A193077 f[x_] := t^(-x); r[n_] := Fibonacci[n];
%t A193077 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193077 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193077 RealDigits[u0, 10]
%Y A193077 Cf. A000045, A001622, A193010, A192232, A193078.
%K A193077 nonn,cons
%O A193077 1,8
%A A193077 _Clark Kimberling_, Jul 15 2011