cp's OEIS Frontend

A193031 Decimal expansion of the constant term of the reduction of 2^x by x^2->x+1.

%I A193031 #10 Jan 19 2022 05:38:14
%S A193031 1,3,1,9,8,7,8,7,2,4,0,2,1,1,5,6,2,7,4,4,5,9,9,7,4,2,1,2,6,3,9,3,1,3,
%T A193031 9,3,1,8,5,9,0,4,4,6,2,3,0,5,5,5,9,7,8,8,1,5,1,7,5,9,4,3,2,8,8,5,3,2,
%U A193031 2,7,6,2,6,1,5,1,5,6,0,3,7,5,5,3,5,6,1,2,4,8,2,3,1,1,3,2,2,0,2,8
%N A193031 Decimal expansion of the constant term of the reduction of 2^x by x^2->x+1.
%C A193031 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193031 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193031 Equals 1 + Sum_{k>=1} log(2)^k*Fibonacci(k-1)/k!.
%F A193031 Equals (1 + (3+4^phi)/sqrt(5))/(phi*2^phi), where phi is the golden ratio (A001622). (End)
%e A193031 1.3198787240211562744599742126393139318590...
%t A193031 f[x_] := 2^x; r[n_] := Fibonacci[n];
%t A193031 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193031 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193031 RealDigits[u0, 10]
%Y A193031 Cf. A000045, A001622, A193010, A192232, A193032.
%K A193031 nonn,cons
%O A193031 1,2
%A A193031 _Clark Kimberling_, Jul 14 2011