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%I #13 Mar 21 2023 07:14:08
%S 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,
%T 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,
%U 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
%N A Jacobsthal ratio.
%C 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
%H G. C. Greubel, <a href="/A084642/b084642.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,1).
%F a(n) = floor(A001045(n+2)/A001045(n+1)).
%F a(n) = floor((2^(n+2) - (-1)^(n+2))/(2^(n+1) - (-1)^(n+1))).
%F From _G. C. Greubel_, Mar 20 2023: (Start)
%F a(n) = A000034(n) + [n=1].
%F a(n) = a(n-2), for n > 3, with a(0) = 1, a(1) = 3, a(2) = 1, a(3) = 2.
%F G.f.: (1 + 3*x - x^3)/(1-x^2).
%F E.g.f.: (1/2)*(2*x + 3*exp(x) - exp(-x)). (End)
%t Table[(3-(-1)^n)/2 +Boole[n==1], {n,0,120}] (* _G. C. Greubel_, Mar 20 2023 *)
%o (Magma) [1,3] cat [1+ (n mod 2): n in [2..120]]; // _G. C. Greubel_, Mar 20 2023
%o (SageMath) [1 + (n%2) + int(n==1) for n in range(121)] # _G. C. Greubel_, Mar 20 2023
%Y Cf. A000034, A001045, A040001.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jun 08 2003
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