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

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, 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, 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

Offset

0,1

Comments

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

Links

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

Formula

From Amiram Eldar, Jan 19 2022: (Start)

Equals 1 - A193087.

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

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

Example

0.5187233386849291969930277770388656030...

Mathematica

f[x_] := Sin[x]^2; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
RealDigits[u0, 10]

Crossrefs

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

Sequence in context: A073116 A201525 A269229 * A154310 A193856 A255294

Adjacent sequences: A193086 A193087 A193088 * A193090 A193091 A193092

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved