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A003818
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a(1)=a(2)=1, a(n+1) = (a(n)^3 +1)/a(n-1).
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2
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OFFSET
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1,3
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Waldemar Pompe (pompe(AT)zodiac1.mimuw.edu.pl)
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EXTENSIONS
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STATUS
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approved
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