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

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, 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, 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, 4

Offset

0,1

Comments

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

The positive value +0.2960... is the coefficient of x in the reduction of (sin(x))^2 by x^2->x+1.

Links

Table of n, a(n) for n=0..99.

Formula

From Amiram Eldar, Jan 19 2022: (Start)

Equals Sum_{k>=0} (-1)^k * 2^(2*k-1) * Fibonacci(2*k)/(2*k)!.

Equals -sin(1)*sin(sqrt(5))/sqrt(5). (End)

Example

-0.29606728112980805107973977013969639291652...

Mathematica

f[x_] := Cos[x]^2; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]

Crossrefs

Cf. A000045, A193010, A192232, A193087.

Sequence in context: A076841 A213819 A361013 * A162916 A057273 A236558

Adjacent sequences: A193085 A193086 A193087 * A193089 A193090 A193091

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved