login
A193075
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, 3, 9, 5, 6, 4, 7, 0, 6, 8, 7, 9, 3, 2, 1, 6, 0, 8, 2, 3, 7, 8, 8, 1, 6, 5, 0, 5, 7, 9, 3, 1, 8, 7, 1, 1, 3, 1, 7, 3, 5, 8, 0, 0, 7, 5, 5, 8, 5, 2, 2, 8, 1, 7, 4, 5, 0, 1, 3, 3, 5, 1, 7, 8, 9, 0, 7, 2, 4, 8, 6, 0, 3, 9, 5, 9, 6, 7, 2, 5, 7, 3, 4, 6, 3, 0, 2, 0, 5, 5, 2, 9, 8, 2, 5, 0, 2, 2, 0
OFFSET
1,3
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 1 + Sum_{k>=1} log(phi)^k*Fibonacci(k-1)/k!.
Equals (sqrt(5)*phi^sqrt(5) + phi^4 - 1)/(5*phi^phi). (End)
EXAMPLE
1.13956470687932160823788165057931871131735800...
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
nonn,cons
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved