cp's OEIS Frontend

A193089 Decimal expansion of the constant term of the reduction of (sin(x))^2 by x^2->x+1.

%I A193089 #13 Jan 19 2022 05:37:49
%S A193089 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 A193089 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 A193089 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 A193089 Decimal expansion of the constant term of the reduction of (sin(x))^2 by x^2->x+1.
%C A193089 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193089 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193089 Equals 1 - A193087.
%F A193089 Equals Sum_{k>=1} (-1)^(k+1) * 2^(2*k-1) * Fibonacci(2*k-1)/(2*k)!.
%F A193089 Equals (1 + 2*phi^2*sin(phi-1)^2 - cos(2*phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)
%e A193089 0.5187233386849291969930277770388656030...
%t A193089 f[x_] := Sin[x]^2; r[n_] := Fibonacci[n];
%t A193089 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193089 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193089 RealDigits[u0, 10]
%Y A193089 Cf. A000045, A001622, A193010, A192232, A193087, A193088.
%K A193089 nonn,cons
%O A193089 0,1
%A A193089 _Clark Kimberling_, Jul 15 2011