A193009 Decimal expansion of the constant term of the reduction of 2^(-x) by x^2->x+1.

1, 2, 0, 0, 6, 2, 2, 0, 0, 3, 9, 6, 9, 0, 5, 8, 5, 2, 1, 0, 0, 7, 3, 9, 7, 6, 5, 9, 0, 4, 3, 8, 7, 8, 2, 6, 7, 1, 7, 8, 0, 5, 5, 0, 0, 2, 6, 3, 9, 4, 8, 2, 1, 8, 6, 6, 5, 1, 2, 1, 9, 2, 1, 0, 4, 2, 7, 7, 3, 8, 0, 7, 9, 4, 3, 8, 0, 8, 3, 4, 8, 6, 2, 5, 3, 9, 2, 5, 3, 7, 1, 6, 6, 4, 3, 8, 8, 3, 5, 1

Offset: 1

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn
• cons

Comments

Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

Examples

			1.20062200396905852100739765904387826717805500...
		

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193035.

Programs

Mathematica

f[x_] := 2^(-x); r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
RealDigits[u0, 10]

Formula

From Amiram Eldar, Jan 19 2022: (Start)

Equals 1 + Sum_{k>=1} (-log(2))^k*Fibonacci(k-1)/k!.

Equals (4^(phi-1)*(5+3*sqrt(5)) + sqrt(5))/(5*phi*2^phi), where phi is the golden ratio (A001622). (End)