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

1, 1, 2, 9, 365, 5403014, 432130991537958813, 14935169284101525874491673463268414536523593057

Offset: 1

Author

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

Keywords

• nonn

Comments

The term a(9) has 121 digits. - Harvey P. Dale, Nov 23 2013

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(n-1) * 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 204-205 (1978).

Links

• British Mathematical Olympiad, 1978 - Problem 3.
• math110, A similar Somos sequence problem prove A_2nB_{n+3} is integer sequence.
• Index to sequences related to Olympiads.

Programs

Maple

A003818 := proc(n) option remember; if n <= 2 then 1 else (A003818(n-1)^3+1)/A003818(n-2); fi; end;

Mathematica

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 *)

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 */

Formula

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

May be extended to negative arguments by setting a(n) = a(3-n) for all n in Z. - Michael Somos, Apr 11 2017

Extensions

More terms from Benoit Cloitre, May 31 2005