A385414 Number of distinct states of Conway's Game of Life, starting from an n-th level Hilbert curve on a toroidal 2^(n+1)-1 by 2^(n+1)-1 grid.

2, 2, 3, 24, 70, 584, 1325, 2082, 5304, 6327, 10679, 11822

Offset: 0

Author

Luke Bennet, Jun 27 2025

Keywords

• nonn
• more

Comments

The curve is taken with segments of length 2 so that it follows a path through coordinates (2*A059252(t), 2*A059253(t)) for 0 <= t < 2^n.

The size of the toroidal grid is the extent of these coordinates so that the cells on one edge are immediately adjacent to the cells on the opposite side.

The grid has a fixed position and orientation and states differing at any cell are distinct.

Examples

			For n=0, the curve is a single cell on a 1 X 1 toroidal grid and has a(0) = 2 states: initially live, then dead and remaining so.
For n=2 the initial state and two subsequent states are
  o o o . o o o | . . . . . . . | . . . . . . . |
  o . o . o . o | . . . . . . . | . . . . . . . |
  o . o o o . o | . . o . o . . | . . . . . . . |
  o . . . . . o | . . . . . . . | . . . . . . . |
  o o o . o o o | . . o . o . . | . . . . . . . |
  . . o . o . . | . . . . . . . | . . . . . . . |
  o o o . o o o | . . . . . . . | . . . . . . . |
  (generation 1)  (generation 2)  (generation 3)
Every generation after 3 is identical to generation 3, so this sequence has 3 distinct states. Thus, a(2) = 3.
		

Crossrefs

Cf. A059252, A059253.