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

1,3

Comments

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

Links

Table of n, a(n) for n=1..100.

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)

Example

1.13956470687932160823788165057931871131735800...

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]

Crossrefs

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

Sequence in context: A092041 A317904 A320960 * A285264 A106586 A010633

Adjacent sequences: A193072 A193073 A193074 * A193076 A193077 A193078

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved