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

%I #10 Jan 18 2022 02:31:29

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

%C 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 From _Amiram Eldar_, Jan 18 2022: (Start)

%F Equals 1 + Sum_{k>=1} (-1)^k*Fibonacci(k-1)/k!.

%F Equals exp(-1/2)*(1 + sqrt(5)/5 + 2/(exp(sqrt(5))-1))*sinh(sqrt(5)/2). (End)

%e 1.39729651650004415809334932390899486052640...

%t f[x_] := Exp[-x]; r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]

%t RealDigits[u0, 10]

%Y Cf. A000045, A193010, A192232, A099935.

%K nonn,cons

%O 1,2

%A _Clark Kimberling_, Jul 14 2011