cp's OEIS Frontend

A193076 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).

%I A193076 #15 Feb 08 2025 03:42:14
%S A193076 6,4,2,0,7,1,0,9,8,8,0,3,6,3,7,5,7,2,2,6,6,3,4,8,4,4,9,3,1,8,3,9,6,9,
%T A193076 4,3,3,2,2,0,8,2,5,3,9,2,8,3,1,8,6,9,4,0,5,9,1,6,5,8,2,9,6,1,5,7,0,9,
%U A193076 5,8,3,5,1,0,6,7,8,9,3,9,4,9,9,7,6,4,1,8,3,3,9,7,8,4,5,2,2,8,9,0,7
%N A193076 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).
%C A193076 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193076 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193076 Equals Sum_{k>=0} log(phi)^k*Fibonacci(k)/k!.
%F A193076 Equals (phi^phi - phi^(1-phi))/sqrt(5). (End)
%e A193076 0.6420710988036375722663484493183969433220...
%t A193076 t = GoldenRatio
%t A193076 f[x_] := t^(x); r[n_] := Fibonacci[n];
%t A193076 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193076 u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t A193076 RealDigits[u1, 10]
%Y A193076 Cf. A000045, A001622, A193010, A192232, A193075.
%K A193076 nonn,cons
%O A193076 0,1
%A A193076 _Clark Kimberling_, Jul 15 2011
%E A193076 a(99)-a(100) from _Georg Fischer_, Feb 08 2025