A193013 Decimal expansion of the constant term of the reduction of cos(x) by x^2->x+1.

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

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 18 2022: (Start)

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

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

Example

0.5767021231946339181854043477561375348200573480...

Mathematica

f[x_] := Cos[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]

Crossrefs

Cf. A000045, A193010, A192232, A193014.

Sequence in context: A346590 A100554 A258148 * A220607 A348607 A200097

Adjacent sequences: A193010 A193011 A193012 * A193014 A193015 A193016

Keyword

nonn,cons

Author

Clark Kimberling, Jul 14 2011

Status

approved