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A084642
A Jacobsthal ratio.
1
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
Decimal expansion of 433/3300. - Elmo R. Oliveira, May 06 2024
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jun 08 2003
STATUS
approved