cp's OEIS Frontend

A193075 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 A193075 #15 Jan 19 2022 05:38:06
%S A193075 1,1,3,9,5,6,4,7,0,6,8,7,9,3,2,1,6,0,8,2,3,7,8,8,1,6,5,0,5,7,9,3,1,8,
%T A193075 7,1,1,3,1,7,3,5,8,0,0,7,5,5,8,5,2,2,8,1,7,4,5,0,1,3,3,5,1,7,8,9,0,7,
%U A193075 2,4,8,6,0,3,9,5,9,6,7,2,5,7,3,4,6,3,0,2,0,5,5,2,9,8,2,5,0,2,2,0
%N A193075 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 A193075 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193075 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193075 Equals 1 + Sum_{k>=1} log(phi)^k*Fibonacci(k-1)/k!.
%F A193075 Equals (sqrt(5)*phi^sqrt(5) + phi^4 - 1)/(5*phi^phi). (End)
%e A193075 1.13956470687932160823788165057931871131735800...
%t A193075 t = GoldenRatio
%t A193075 f[x_] := t^(x); r[n_] := Fibonacci[n];
%t A193075 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193075 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193075 RealDigits[u0, 10]
%Y A193075 Cf. A000045, A001622, A193010, A192232, A193076.
%K A193075 nonn,cons
%O A193075 1,3
%A A193075 _Clark Kimberling_, Jul 15 2011