A373566 Expansion of x - 1/(x - 1/(x + 1)).

1, 3, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155

Offset

0,2

Links

Table of n, a(n) for n=0..38.

Index entries for linear recurrences with constant coefficients, signature (1,1).

Formula

a(n) = [x^n] (x^3 + x^2 - 2*x - 1) / (x^2 + x - 1).

From Stefano Spezia, Jun 10 2024: (Start)

a(n) = 2^(-n-1)*((1 - sqrt(5))^n*(sqrt(5) - 3) + (1 + sqrt(5))^n*(sqrt(5) + 3))/sqrt(5) for n <> 1.

E.g.f.: x + exp(x/2)*(5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2))/5. (End)

Mathematica

CoefficientList[Series[x - 1/(x - 1/(x + 1)), {x, 0, 38}], x] (* Michael De Vlieger, Jun 10 2024 *)

Crossrefs

Essentially the same as A071679 and A020695.

Sequence in context: A200737 A200741 A271970 * A213678 A293676 A303714

Adjacent sequences: A373563 A373564 A373565 * A373567 A373568 A373569

Keyword

nonn,easy

Author

Peter Luschny, Jun 10 2024

Status

approved