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%I #15 Jun 01 2018 01:55:19
%S 1,2,4,7,13,14,20
%N The number of cycles in the Vers de Verres game, where 'worms' are transferred between 'cups' in a deterministic fashion. Because this defines a finite-state automaton, we know that every state eventually enters a cycle (or fixed point, which is essentially a cycle of length 1). The number of 'cups' (frequently called 'n') is a parameter for this automaton, and so we count the cycles (and fixed points) with respect to n.
%C The game is described in the websites listed, and already has other sequences, e.g., A151986. Note that this also gives the number of connected components, if we draw a graph of this process. The sequence gives the number of cycles, for a given number of cups. The sequence is increasing (append a 0 to all configurations in a cycle, and you get the same cycle with one more cup). It is strictly increasing since {n-1,0,0,0...,0} occurs in a cycle at stage n, but never before.
%C I am not clear on how this is meant to differ from A176450; my calculations reproduce the terms there not the ones in this sequence. - _Joseph Myers_, Nov 13 2010
%H Eric Angelini - <a href="http://www.cetteadressecomportecinquantesignes.com/GlassWorms.htm">Vers de Verres</a>
%H E. Angelini, <a href="/A151986/a151986.pdf">Vers de verres (Glass worms)</a> [Cached copy, with permission]
%H Kellen Myers - <a href="http://math.rutgers.edu/~kellenm/ExpMath/worms.html">Vers de Verres</a> [Broken link]
%e For n=4, there are seven cycles: {0300,3000,0030}, {3300,3003,0330}, {0200,2000}, {3330}, {2200}, {1000}, {0000}. Note that four of these are "inherited" from n=3, as described above.
%Y Related to A151986, A151987, A176336.
%K more,nonn,obsc,uned
%O 1,2
%A _Kellen Myers_, May 02 2010
%E Fixed error in sequence. Added small amount of formatting changes and elaboration. - _Kellen Myers_, May 03 2010
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