%I #11 Jan 19 2022 05:37:58

%S 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 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 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 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 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 1 + Sum_{k>=1} (-log(phi))^k*Fibonacci(k-1)/k!.

%F Equals (1 + phi^(2*phi+1))/(sqrt(5)*phi^(phi+1)). (End)

%e 1.101111483485871838026720619840...

%t t = GoldenRatio

%t f[x_] := t^(-x); r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]

%t RealDigits[u0, 10]

%Y Cf. A000045, A001622, A193010, A192232, A193078.

%K nonn,cons

%O 1,8

%A _Clark Kimberling_, Jul 15 2011