A193012 Decimal expansion of the coefficient of x in the reduction of sin(x) by x^2->x+1.

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

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

			0.705845470457234393737808753502039387530431888887345001...
		

Crossrefs

Cf. A000045, A193010, A193011, A192232.

Programs

Mathematica

f[x_] := Sin[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 200}], 100]
RealDigits[u1, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)

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

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