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

Author

Paul Barry, Jun 08 2003

Keywords

• easy
• nonn

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

Decimal expansion of 433/3300. - Elmo R. Oliveira, May 06 2024

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1).

Crossrefs

Cf. A000034, A001045, A040001.

Programs

Magma

[1,3] cat [1+ (n mod 2): n in [2..120]]; // G. C. Greubel, Mar 20 2023

Mathematica

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

SageMath

[1 + (n%2) + int(n==1) for n in range(121)] # G. C. Greubel, Mar 20 2023

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)