A022405 - OEIS

%I #45 Sep 08 2022 08:44:46

%S 0,1,2,2,3,4,10,37,366,13532,4952675,67019597734,331926286207224918,

%T 22245566178948766568816183137,

%U 7383888166355511098764350563784314022618210032

%N a(n) = a(n-1)*a(n-2) - a(n-3), with a(1) = 0, a(2) = 1, a(3) = 2.

%C Theorem 1.1 of Hare et al. (2010, 2011) involves a shifted version of this sequence and the Fibonacci sequence A000045. (The program by Alonso del Arte below does involve a shifted version of this sequence.) - _Petros Hadjicostas_, May 11 2019

%H G. C. Greubel, <a href="/A022405/b022405.txt">Table of n, a(n) for n = 1..21</a>

%H Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, <a href="http://arxiv.org/abs/1006.2117">An explicit counterexample to the Lagarias-Wang finiteness conjecture</a>, arXiv:1006.2117 [math.OC], 2010-2011.

%H Kevin G. Hare, Ian D. Morris, Nikita Sidorov, and Jacques Theys, <a href="https://doi.org/10.1016/j.aim.2010.12.012">An explicit counterexample to the Lagarias-Wang finiteness conjecture</a>, Advances in Mathematics 226 (2011), 4667-4701.

%F It appears that lim_{n->infinity} log(a(n))/phi^n = 0.07743008049000107520747623421744398272089261907514..., where phi = (1 + sqrt(5))/2 is the golden ratio A001622. - _Petros Hadjicostas_ and _Jon E. Schoenfield_, May 11 2019

%t a[1] = 0; a[2] = 1; a[3] = 2; a[n_] := a[n] = a[n - 1] a[n - 2] - a[n - 3]; Table[a[n], {n, 1, 15}] (* _Alonso del Arte_, Jan 31 2011 *)

%o (Magma) I:=[0,1,2]; [n le 3 select I[n] else Self(n-1)*Self(n-2) - Self(n-3): n in [1..15]];  // _G. C. Greubel_, Mar 01 2018

%Y Cf. A001622, A061021, A061292, A072878, A072879, A072880, A178768.

%K easy,nonn

%O 1,3

%A _Robert G. Wilson v_, Jul 05 2000

%E Name clarified by _Michel Marcus_, May 10 2019