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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
From Amiram Eldar, Jan 19 2022: (Start)
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
Sequence in context: A309217 A217602 A300690 * A271871 A204993 A234001
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)