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a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.
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%I #57 May 16 2019 08:50:25

%S 1,1,1,1,3,11,131,17291,99665321,903016046275353,

%T 6224717403288400029624460201,

%U 2240882930472585840954332388399544581477407095086361,50384188378657848181032338163962292285660644698840136656562636145266593550842871302412156442811

%N a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4) with a(1) = a(2) = a(3) = a(4) = 1.

%C A subsequence of the generalized Markoff numbers.

%H Seiichi Manyama, <a href="/A072878/b072878.txt">Table of n, a(n) for n = 1..16</a>

%H Arthur Baragar, <a href="http://dx.doi.org/10.1006/jnth.1994.1078">Integral solutions of the Markoff-Hurwitz equations</a>, J. Number Theory 49 (1994), 27-44.

%H Andrew N. W. Hone, <a href="https://arxiv.org/abs/math/0601324">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, arXiv:math/0601324 [math.NT], 2006.

%H Andrew N. W. Hone, <a href="https://doi.org/10.1088/0305-4470/39/12/L01">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, J. Phys. A: Math. Gen. 39 (2006), L171-L177.

%H Matthew Christopher Russell, <a href="https://doi.org/doi:10.7282/T3MC926D">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrence</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

%F a(1) = a(2) = a(3) = a(4) = 1; a(n) = (a(n-1)^2 + a(n-3)^2 + a(n-2)^2)/a(n-4) for n >= 5.

%F From the recurrence a(n) = 4*a(n-1)*a(n-2)*a(n-3) - a(n-4), any four successive terms satisfy the Markoff-Hurwitz equation a(n)^2 + a(n-1)^2 + a(n-2)^2 + a(n-3)^2 = 4*a(n)*a(n-1)*a(n-2)*a(n-3), cf. A075276. As n tends to infinity, the limit of log(log(a(n)))/n is log x = 0.6093778633..., where x=1.839286755... is the real root of the cubic x^3 - x^2 - x - 1 = 0. - _Andrew Hone_, Nov 14 2005

%t RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==4a[n-1]a[n-2]a[n-3]-a[n-4]},a,{n,15}] (* _Harvey P. Dale_, Nov 29 2014 *)

%Y Cf. A022405, A064098, A075276, A072879, A072880.

%K easy,nonn

%O 1,5

%A _Benoit Cloitre_, Jul 28 2002

%E Entry revised Nov 19 2005, based on comments from _Andrew Hone_

%E a(13) from _Harvey P. Dale_, Nov 29 2014

%E Name clarified by _Petros Hadjicostas_, May 11 2019