The online magazine, Quantamagazine, recently had an article
on special recursive sequences: https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/.
The family of sequences I found most interesting was the Somos-k sequences. They
are named after Michael Somos who discovered them in the 1980s.
For example, terms in the Somos-4 sequence are given by a(0)=1,
a(1)=1, a(2)=1, a(3)=1 and for n>3,
a(n)= (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4). So, the Somos-4 sequence is: 1, 1,
1, 1, 2, 3, 7, 23, 59, 314, 1529, 8209…
The interesting thing about this sequence is that all the
terms are integers even though there is a divisor in the equation of a(n). This
is true for all Somos-k sequences with k<8. Once k exceeds 8, the sequences
start with integers but eventually yield non-integer values. With k=8, this
occurs with 18th term.
A Numberphile video speaks about this specific
sequence, called the Troublemaker Number: https://www.youtube.com/watch?v=p-HN_ICaCyM.
Somos-related sequences in the Online Encyclopedia of Integer
Sequences include:
Somos-4: https://oeis.org/A006720
Somos-5: https://oeis.org/A006721
Somos-6: https://oeis.org/A006722
Somos-7: https://oeis.org/A006723
Term at which n Somos-k sequence first becomes nonintegral
(for k>7): https://oeis.org/A030127
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