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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).
2
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.
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
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved