|
|
A193077
|
|
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).
|
|
2
|
|
|
1, 1, 0, 1, 1, 1, 1, 4, 8, 3, 4, 8, 5, 8, 7, 1, 8, 3, 8, 0, 2, 6, 7, 2, 0, 6, 1, 9, 8, 4, 0, 9, 9, 7, 5, 8, 1, 1, 9, 0, 2, 8, 5, 1, 1, 9, 0, 3, 3, 6, 2, 5, 4, 5, 1, 7, 2, 5, 8, 3, 9, 6, 4, 1, 3, 8, 0, 7, 6, 5, 2, 2, 9, 5, 6, 0, 0, 1, 7, 8, 1, 3, 5, 3, 1, 8, 5, 1, 7, 9, 8, 7, 6, 8, 4, 1, 5, 9, 0, 0
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,8
|
|
COMMENTS
|
Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
|
|
LINKS
|
|
|
FORMULA
|
Equals 1 + Sum_{k>=1} (-log(phi))^k*Fibonacci(k-1)/k!.
Equals (1 + phi^(2*phi+1))/(sqrt(5)*phi^(phi+1)). (End)
|
|
EXAMPLE
|
1.101111483485871838026720619840...
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|