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

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, 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, 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

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 - A193089.

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

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

Example

0.48127666131507080300697222296113439695...

Mathematica

f[x_] := Cos[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, A193088, A193089.

Sequence in context: A154520 A244690 A002390 * A201404 A122149 A254149

Adjacent sequences: A193084 A193085 A193086 * A193088 A193089 A193090

Keyword

nonn,cons

Author

Clark Kimberling, Jul 15 2011

Status

approved