A193078 Decimal expansion of the coefficient of x in the reduction of phi^(-x) by x^2->x+1, where phi=(1+sqrt(5))/2 is the golden ratio (A001622) (negated).

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

Offset

0,1

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=0..99.

Formula

From Amiram Eldar, Jan 19 2022: (Start)

Equals Sum_{k>=0} (-log(phi))^k*Fibonacci(k)/k!.

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

Example

-0.39682176225463996686831560297353019716760...

Mathematica

t = GoldenRatio
f[x_] := t^(-x); r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193077.

Sequence in context: A153416 A269226 A205557 * A021256 A349850 A331063

Adjacent sequences: A193075 A193076 A193077 * A193079 A193080 A193081

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved