cp's OEIS Frontend

A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.

%I A193026 #10 Jan 18 2022 02:31:29
%S A193026 1,3,9,7,2,9,6,5,1,6,5,0,0,0,4,4,1,5,8,0,9,3,3,4,9,3,2,3,9,0,8,9,9,4,
%T A193026 8,6,0,5,2,6,4,0,8,7,4,3,7,2,3,7,0,9,2,3,3,5,6,4,0,8,2,8,9,0,2,5,9,3,
%U A193026 6,7,5,9,2,4,7,1,6,5,8,7,6,7,5,3,6,4,1,3,7,5,5,7,8,3,4,4,0,2,4,3
%N A193026 Decimal expansion of the constant term of the reduction of e^(-x) by x^2->x+1.
%C A193026 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.  The coefficient of x in this reduction is the constant at A099935.
%F A193026 From _Amiram Eldar_, Jan 18 2022: (Start)
%F A193026 Equals 1 + Sum_{k>=1} (-1)^k*Fibonacci(k-1)/k!.
%F A193026 Equals exp(-1/2)*(1 + sqrt(5)/5 + 2/(exp(sqrt(5))-1))*sinh(sqrt(5)/2). (End)
%e A193026 1.39729651650004415809334932390899486052640...
%t A193026 f[x_] := Exp[-x]; r[n_] := Fibonacci[n];
%t A193026 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193026 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193026 RealDigits[u0, 10]
%Y A193026 Cf. A000045, A193010, A192232, A099935.
%K A193026 nonn,cons
%O A193026 1,2
%A A193026 _Clark Kimberling_, Jul 14 2011