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A072878 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. 14
1, 1, 1, 1, 3, 11, 131, 17291, 99665321, 903016046275353, 6224717403288400029624460201, 2240882930472585840954332388399544581477407095086361, 50384188378657848181032338163962292285660644698840136656562636145266593550842871302412156442811 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

A subsequence of the generalized Markoff numbers.

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 recurrence, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

FORMULA

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.

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

MATHEMATICA

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 *)

CROSSREFS

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

Sequence in context: A088076 A276258 A284604 * A112957 A057205 A121897

Adjacent sequences:  A072875 A072876 A072877 * A072879 A072880 A072881

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Jul 28 2002

EXTENSIONS

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

a(13) from Harvey P. Dale, Nov 29 2014

Name clarified by Petros Hadjicostas, May 11 2019

STATUS

approved

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Last modified November 22 10:59 EST 2019. Contains 329389 sequences. (Running on oeis4.)