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.
1, 1, 1, 1, 3, 11, 131, 17291, 99665321, 903016046275353, 6224717403288400029624460201, 2240882930472585840954332388399544581477407095086361, 50384188378657848181032338163962292285660644698840136656562636145266593550842871302412156442811
Offset: 1
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.
Programs
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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 *)
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
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
Comments