A177101 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.

1, 2, 4, 7, 13, 14, 20

Offset: 1

Author

Kellen Myers, May 02 2010

Keywords

• more
• nonn
• obsc
• uned

Comments

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.

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

Examples

			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.
		

Links

• Eric Angelini - Vers de Verres
• E. Angelini, Vers de verres (Glass worms) [Cached copy, with permission]
• Kellen Myers - Vers de Verres [Broken link]

Crossrefs

Related to A151986, A151987, A176336.

Extensions

Fixed error in sequence. Added small amount of formatting changes and elaboration. - Kellen Myers, May 03 2010