A193089 - OEIS

%I #13 Jan 19 2022 05:37:49

%S 5,1,8,7,2,3,3,3,8,6,8,4,9,2,9,1,9,6,9,9,3,0,2,7,7,7,7,0,3,8,8,6,5,6,

%T 0,3,0,4,1,5,2,8,4,0,5,3,5,4,2,4,5,3,7,9,2,7,2,3,3,3,6,9,3,7,4,6,9,9,

%U 7,2,0,6,8,7,5,2,5,5,6,0,8,3,1,2,8,2,1,0,5,5,2,4,0,1,3,1,5,4,7,0

%N Decimal expansion of the constant term of the reduction of (sin(x))^2 by x^2->x+1.

%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 1 - A193087.

%F Equals Sum_{k>=1} (-1)^(k+1) * 2^(2*k-1) * Fibonacci(2*k-1)/(2*k)!.

%F Equals (1 + 2*phi^2*sin(phi-1)^2 - cos(2*phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)

%e 0.5187233386849291969930277770388656030...

%t f[x_] := Sin[x]^2; r[n_] := Fibonacci[n];

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

%t u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]

%t RealDigits[u0, 10]

%Y Cf. A000045, A001622, A193010, A192232, A193087, A193088.

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Jul 15 2011