cp's OEIS Frontend

A193088 Decimal expansion of the coefficient of x in the reduction of (cos(x))^2 by x^2->x+1 (negated).

%I A193088 #19 Feb 08 2025 03:42:09
%S A193088 2,9,6,0,6,7,2,8,1,1,2,9,8,0,8,0,5,1,0,7,9,7,3,9,7,7,0,1,3,9,6,9,6,3,
%T A193088 9,2,9,1,6,5,2,7,5,6,4,7,6,9,2,8,5,8,3,3,3,5,0,7,5,8,0,7,9,8,7,6,2,8,
%U A193088 9,2,5,3,0,4,5,2,5,8,2,4,8,8,3,8,4,4,3,4,7,1,8,5,1,6,5,6,5,7,3,3,9
%N A193088 Decimal expansion of the coefficient of x in the reduction of (cos(x))^2 by x^2->x+1 (negated).
%C A193088 Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
%C A193088 The positive value +0.2960... is the coefficient of x in the reduction of (sin(x))^2 by x^2->x+1.
%F A193088 From _Amiram Eldar_, Jan 19 2022: (Start)
%F A193088 Equals Sum_{k>=0} (-1)^k * 2^(2*k-1) * Fibonacci(2*k)/(2*k)!.
%F A193088 Equals -sin(1)*sin(sqrt(5))/sqrt(5). (End)
%e A193088 -0.29606728112980805107973977013969639291652...
%t A193088 f[x_] := Cos[x]^2; r[n_] := Fibonacci[n];
%t A193088 c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
%t A193088 u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
%t A193088 RealDigits[u1, 10]
%Y A193088 Cf. A000045, A193010, A192232, A193087.
%K A193088 nonn,cons
%O A193088 0,1
%A A193088 _Clark Kimberling_, Jul 15 2011
%E A193088 a(99)-a(100) from _Georg Fischer_, Feb 08 2025