A193011 Decimal expansion of the constant term of the reduction of sin(x) by x^2->x+1.

1, 4, 3, 1, 9, 7, 4, 5, 2, 9, 1, 0, 0, 8, 0, 2, 3, 2, 6, 4, 1, 5, 0, 3, 6, 4, 7, 7, 2, 4, 6, 4, 1, 0, 8, 4, 9, 9, 0, 8, 8, 9, 2, 8, 6, 2, 3, 8, 9, 1, 3, 5, 5, 1, 7, 4, 6, 3, 3, 8, 8, 6, 4, 3, 3, 5, 0, 1, 0, 7, 1, 2, 6, 6, 1, 3, 3, 6, 4, 0, 8, 9, 0, 9, 5, 6, 1, 9, 7, 3, 1, 0, 8, 5, 2, 8, 5, 9, 1, 0

Offset: 0

Author

Clark Kimberling, Jul 14 2011

Keywords

• nonn
• cons

Comments

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

Examples

			-0.143197452910080232641503647724641084990...
		

Crossrefs

Cf. A000045, A193010, A192232, A193012.

Programs

Mathematica

f[x_] := Sin[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 200}], 100]
RealDigits[u0, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)

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

Equals cos(sqrt(5)/2)*sin(1/2) - cos(1/2)*sin(sqrt(5)/2)/sqrt(5). (End)