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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

%I #12 Jan 19 2022 05:37:39

%S 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,

%T 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,

%U 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

%N 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).

%C Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

%F From _Amiram Eldar_, Jan 19 2022: (Start)

%F Equals Sum_{k>=0} (-log(phi))^k*Fibonacci(k)/k!.

%F Equals -(phi^sqrt(5) - 1)/(sqrt(5)*phi^phi). (End)

%e -0.39682176225463996686831560297353019716760...

%t t = GoldenRatio

%t f[x_] := t^(-x); r[n_] := Fibonacci[n];

%t c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]

%t u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]

%t RealDigits[u1, 10]

%Y Cf. A000045, A001622, A193010, A192232, A193077.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Jul 15 2011