cp's OEIS Frontend

A193087 Decimal expansion of the constant term of the reduction of (cos(x))^2 by x^2->x+1.

%I A193087 #10 Jan 19 2022 05:37:53
%S A193087 4,8,1,2,7,6,6,6,1,3,1,5,0,7,0,8,0,3,0,0,6,9,7,2,2,2,2,9,6,1,1,3,4,3,
%T A193087 9,6,9,5,8,4,7,1,5,9,4,6,4,5,7,5,4,6,2,0,7,2,7,6,6,6,3,0,6,2,5,3,0,0,
%U A193087 2,7,9,3,1,2,4,7,4,4,3,9,1,6,8,7,1,7,8,9,4,4,7,5,9,8,6,8,4,5,3,0
%N A193087 Decimal expansion of the constant term of the reduction of (cos(x))^2 by x^2->x+1.
%C A193087 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%F A193087 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193087 Equals 1 - A193089.
%F A193087 Equals 1 + Sum_{k>=1} (-1)^k * 2^(2*k-1) * Fibonacci(2*k-1)/(2*k)!.
%F A193087 Equals 1 + (cos(2*phi) - 2*phi^2*sin(phi-1)^2 - 1)/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)
%e A193087 0.48127666131507080300697222296113439695...
%t A193087 f[x_] := Cos[x]^2; r[n_] := Fibonacci[n];
%t A193087 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193087 u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
%t A193087 RealDigits[u0, 10]
%Y A193087 Cf. A000045, A001622, A193010, A192232, A193088, A193089.
%K A193087 nonn,cons
%O A193087 0,1
%A A193087 _Clark Kimberling_, Jul 15 2011