

A003818


a(1)=a(2)=1, a(n+1) = (a(n)^3 +1)/a(n1).


2




OFFSET

1,3


COMMENTS

The recursion has the Laurent property. If a(1), a(2) are variables, then a(n) is a Laurent polynomial (a rational function with a monomial denominator).  Michael Somos, Feb 25 2019
This sequence was the subject of the 3rd problem of the 14th British Mathematical Olympiad in 1978 where this sequence was defined by: u(1) = 1, u(1) < u(2) and u(n)^3 + 1 = u(n1) * u(n+1), for n > 1 (see link B. M. O. and reference).  Bernard Schott, Apr 01 2021


REFERENCES

A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 3 pp. 68 and 204205 (1978).


LINKS



FORMULA

a(n) is asymptotic to c^F(2n) where F(n) is the nth Fibonacci's number A000045(n) and c=1.1137378757136...  Benoit Cloitre, May 31 2005
May be extended to negative arguments by setting a(n) = a(3n) for all n in Z.  Michael Somos, Apr 11 2017


MAPLE



MATHEMATICA

RecurrenceTable[{a[1]==a[2]==1, a[n]==(a[n1]^3+1)/a[n2]}, a, {n, 10}] (* Harvey P. Dale, Nov 23 2013 *)


PROG

(PARI) {a(n) = if( n<1, n=3n); if( n<3, 1, (1 + a(n1)^3) / a(n2))}; /* Michael Somos, Apr 11 2017 */


CROSSREFS



KEYWORD

nonn


AUTHOR

Waldemar Pompe (pompe(AT)zodiac1.mimuw.edu.pl)


EXTENSIONS



STATUS

approved



