A084642 A Jacobsthal ratio.

1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1

Offset

0,2

Comments

The Jacobsthal recurrence means that A001045(n+1)/A001045(n) = 1 + 2/(A001045(n)/A001045(n-1)). The sequence of these fractions alternates after the first terms values just above 2 and just below 2, because the mapping x -> 1+2/x is concave in the neighborhood of x=2, where x=2 is an attractor. As a consequence, this sequence here iterates like A040001 or A000034 after a few terms. - R. J. Mathar, Sep 17 2008

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,1).

Formula

a(n) = floor(A001045(n+2)/A001045(n+1)).

a(n) = floor((2^(n+2) - (-1)^(n+2))/(2^(n+1) - (-1)^(n+1))).

From G. C. Greubel, Mar 20 2023: (Start)

a(n) = A000034(n) + [n=1].

a(n) = a(n-2), for n > 3, with a(0) = 1, a(1) = 3, a(2) = 1, a(3) = 2.

G.f.: (1 + 3*x - x^3)/(1-x^2).

E.g.f.: (1/2)*(2*x + 3*exp(x) - exp(-x)). (End)

Mathematica

Table[(3-(-1)^n)/2 +Boole[n==1], {n, 0, 120}] (* G. C. Greubel, Mar 20 2023 *)

Prog

(Magma) [1, 3] cat [1+ (n mod 2): n in [2..120]]; // G. C. Greubel, Mar 20 2023
(SageMath) [1 + (n%2) + int(n==1) for n in range(121)] # G. C. Greubel, Mar 20 2023

Crossrefs

Cf. A000034, A001045, A040001.

Sequence in context: A288818 A319658 A276988 * A271418 A230500 A010281

Adjacent sequences: A084639 A084640 A084641 * A084643 A084644 A084645

Keyword

easy,nonn

Author

Paul Barry, Jun 08 2003

Status

approved