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A003818 a(1)=a(2)=1, a(n+1) = (a(n)^3 +1)/a(n-1).

%I A003818 #27 Apr 01 2021 12:18:10
%S A003818 1,1,2,9,365,5403014,432130991537958813,
%T A003818 14935169284101525874491673463268414536523593057
%N A003818 a(1)=a(2)=1, a(n+1) = (a(n)^3 +1)/a(n-1).
%C A003818 The term a(9) has 121 digits. - _Harvey P. Dale_, Nov 23 2013
%C A003818 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
%C A003818 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(n-1) * u(n+1), for n > 1 (see link B. M. O. and reference). - _Bernard Schott_, Apr 01 2021
%D A003818 A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Pb 3 pp. 68 and 204-205 (1978).
%H A003818 British Mathematical Olympiad, <a href="https://bmos.ukmt.org.uk/home/bmo-1978.pdf">1978 - Problem 3</a>.
%H A003818 math110, <a href="http://math.stackexchange.com/questions/1932492/">A similar Somos sequence problem prove A_2nB_{n+3} is integer sequence</a>.
%H A003818 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%F A003818 a(n) is asymptotic to c^F(2n) where F(n) is the n-th Fibonacci's number A000045(n) and c=1.1137378757136... - _Benoit Cloitre_, May 31 2005
%F A003818 May be extended to negative arguments by setting a(n) = a(3-n) for all n in Z. - _Michael Somos_, Apr 11 2017
%p A003818 A003818 := proc(n) option remember; if n <= 2 then 1 else (A003818(n-1)^3+1)/A003818(n-2); fi; end;
%t A003818 RecurrenceTable[{a[1]==a[2]==1,a[n]==(a[n-1]^3+1)/a[n-2]},a,{n,10}] (* _Harvey P. Dale_, Nov 23 2013 *)
%o A003818 (PARI) {a(n) = if( n<1, n=3-n); if( n<3, 1, (1 + a(n-1)^3) / a(n-2))}; /* _Michael Somos_, Apr 11 2017 */
%K A003818 nonn
%O A003818 1,3
%A A003818 Waldemar Pompe (pompe(AT)zodiac1.mimuw.edu.pl)
%E A003818 More terms from _Benoit Cloitre_, May 31 2005