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A072879 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. 12
1, 1, 1, 1, 1, 4, 19, 379, 144019, 20741616379, 107553662508585672001, 608831069421618273050865038881215685876, 978035016076705458999330010986670207956236476587064788804921180339451725001 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Solutions of the Hurwitz equation in five variables.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 1..16

Arthur Baragar, Integral solutions of the Markoff-Hurwitz equations, J. Number Theory 49 (1994), 27-44.

Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.

Andrew N. W. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, J. Phys. A: Math. Gen. 39 (2006), L171-L177.

Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

FORMULA

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.

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

MATHEMATICA

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 *)

CROSSREFS

Cf. A006720, A064098, A072878, A072880.

Sequence in context: A126147 A007411 A276259 * A112958 A080991 A000844

Adjacent sequences:  A072876 A072877 A072878 * A072880 A072881 A072882

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Jul 28 2002

EXTENSIONS

Entry revised Nov 19 2005, based on comments from Andrew Hone

Name clarified by Petros Hadjicostas, May 11 2019

STATUS

approved

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Last modified February 26 21:58 EST 2020. Contains 332295 sequences. (Running on oeis4.)