A193075 Decimal expansion of the constant term of the reduction of phi^x by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622).

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

Offset: 1

Author

Clark Kimberling, Jul 15 2011

Keywords

• nonn
• cons

Comments

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

Examples

			1.13956470687932160823788165057931871131735800...
		

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193076.

Programs

Mathematica

t = GoldenRatio
f[x_] := t^(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(phi)^k*Fibonacci(k-1)/k!.

Equals (sqrt(5)*phi^sqrt(5) + phi^4 - 1)/(5*phi^phi). (End)