|
%I #49 May 16 2019 08:50:31
%S 1,1,1,1,1,4,19,379,144019,20741616379,107553662508585672001,
%T 608831069421618273050865038881215685876,
%U 978035016076705458999330010986670207956236476587064788804921180339451725001
%N a(n) = 5*a(n-1)*a(n-2)*a(n-3)*a(n-4) - a(n-5) with a(1) = a(2) = a(3) = a(4) = a(5) = 1.
%C Solutions of the Hurwitz equation in five variables.
%H Seiichi Manyama, <a href="/A072879/b072879.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 recurrences</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
%F a(1) = a(2) = a(3) = a(4) = a(5) = 1; a(n) = (a(n-1)^2+a(n-2)^2+a(n-3)^2+a(n-4)^2)/a(n-5) for n >= 6.
%F From the recurrence a(n) = 5*a(n-1)*a(n-2)*a(n-3)*a(n-4) - a(n-5), any five successive terms satisfy the five-variable Hurwitz equation a(n)^2+a(n-1)^2+a(n-2)^2+a(n-3)^2+a(n-4)^2 = 5*a(n)*a(n-1)*a(n-2)*a(n-3)*a(n-4). As n tends to infinity, the limit of log(log(a(n)))/n is log x = 0.6562559790..., where x=1.927561975... is the largest real root of the quartic x^4-x^3-x^2-x-1=0. - _Andrew Hone_, Nov 16 2005
%t nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,(5b c d e)-a}; NestList[nxt,{1,1,1,1,1},20][[All,1]] (* _Harvey P. Dale_, Nov 07 2016 *)
%Y Cf. A006720, A064098, A072878, A072880.
%K easy,nonn
%O 1,6
%A _Benoit Cloitre_, Jul 28 2002
%E Entry revised Nov 19 2005, based on comments from _Andrew Hone_
%E Name clarified by _Petros Hadjicostas_, May 11 2019
|
