A193081 Decimal expansion of the constant term of the reduction of cosh(2x) by x^2->x+1.

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

Offset: 1

Author

Clark Kimberling, Jul 15 2011

Keywords

• nonn
• cons

Comments

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

Examples

			4.8707186334008057368566594211377757838921196937...
		

Crossrefs

Cf. A000045, A193010, A192232, A193082.

Programs

Mathematica

f[x_] := Cosh[2 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, 100}], 100]
RealDigits[u0, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)

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

Equals ((3+sqrt(5))*cosh(1-sqrt(5)) + 2*cosh(1+sqrt(5)))/(5 + sqrt(5)). (End)