cp's OEIS Frontend

A193034 Decimal expansion of the coefficient of x in the reduction of 3^x by x^2->x+1.

%I A193034 #12 Jan 19 2022 05:26:42
%S A193034 2,4,1,8,7,3,7,0,6,7,1,8,9,5,5,7,2,5,6,1,4,6,3,7,2,6,6,0,4,3,0,0,2,8,
%T A193034 4,4,5,9,4,3,0,2,6,0,9,1,7,6,7,7,4,1,6,4,7,4,7,0,9,9,7,1,2,6,9,3,4,4,
%U A193034 9,5,6,7,1,6,2,8,1,7,4,6,6,5,9,4,8,1,4,1,3,9,9,0,9,2,3,0,7,8,6,6
%N A193034 Decimal expansion of the coefficient of x in the reduction of 3^x by x^2->x+1.
%C A193034 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193034 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193034 Equals Sum_{k>=0} log(3)^k*Fibonacci(k)/k!.
%F A193034 Equals (3^sqrt(5)-1)/(sqrt(5)*3^(phi-1)), where phi is the golden ratio (A001622). (End)
%e A193034 2.4187370671895572561463726604300...
%t A193034 f[x_] := 3^x; r[n_] := Fibonacci[n];
%t A193034 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193034 u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t A193034 RealDigits[u1, 10]
%Y A193034 Cf. A000045, A001622, A193010, A192232, A193033.
%K A193034 nonn,cons
%O A193034 1,1
%A A193034 _Clark Kimberling_, Jul 14 2011