A193017 Decimal expansion of the constant term of the reduction of cosh(x) by x^2->x+1.

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

Offset: 1

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

			1.5906097564064614629697931199995829165183536750...
		

Crossrefs

Cf. A000045, A001622, A193010, A192232, A193025.

Programs

Mathematica

f[x_] := Cosh[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, 300}], 100]
RealDigits[u0, 10]

Formula

From Amiram Eldar, Jan 18 2022: (Start)

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

Equals (2 * cosh(phi) + (3+sqrt(5))*cosh(1/phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)