A193076 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).

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

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

Example

0.6420710988036375722663484493183969433220...

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, A193075.

Sequence in context: A211022 A021613 A157701 * A073449 A134300 A226655

Adjacent sequences: A193073 A193074 A193075 * A193077 A193078 A193079

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved