A196447 The number of parents of successive approximations used in a greedy approach to creating a Garden of Eden in Conway's Game of Life.

140, 417, 1164, 1005, 3141, 2835, 8797, 7918, 7268, 23415, 21576, 20648, 65342, 62390, 60038, 59165, 177559, 158105, 144487, 136744, 398009, 345711, 317176, 293203, 256688, 822470, 760976, 731808, 714462, 650945, 2087659, 1914317, 1818736, 1811165, 1670837

Offset

1,1

Comments

In order to create a Garden of Eden (i.e., a pattern with no parents) in Conway's Game of Life, we greedily proceed as follows:

Consider one cell. If it is ON it has 140 parents, and if it is OFF it has 372 parents. Therefore we set it ON so as to have a smaller number of parents.

Then we consider an adjacent cell. If it is ON then the two-cell pattern has 417 parents, and if it is OFF then it has 703 parents. Therefore we set it ON so as to have a smaller number of parents.

We continue in this way, considering cells one at a time in a pattern that spirals around the starting cell. For each cell, we choose it to be ON or OFF based on which of those options results in the pattern having fewer parents.

This algorithm eventually produces a Garden of Eden -- that is, a pattern with no parents. This happens when adding cell 266. This sequence is the numbers of parents at each stage during creation of such a Garden of Eden.

All OFF cell numbers are enumerated in sequence A197734.

The Garden of Eden created by this algorithm does not have the minimum possible number of cells, but it is easier to understand and create than most other Gardens of Eden.

Links

Nicolay Beluchenko, Table of n, a(n) for n = 1..266 (complete sequence)

Randall D. Beer, Cultivating the Garden of Eden, arXiv:2210.07837 [nlin.CG], 2022.

LifeWiki, Garden of Eden

Wikipedia, Garden of Eden

Example

a(1) = 140 because the one-cell configuration produced by this algorithm (with one ON cell) has 140 parents
a(2) = 417 because the two-cell configuration produced by this algorithm (with both cells ON) has 417 parents
a(266) = 0 because the 266-cell configuration produced by this algorithm has 0 parents (i.e., it is a Garden of Eden)

Crossrefs

Cf. A197734.

Sequence in context: A337689 A360217 A249554 * A255783 A255776 A242319

Adjacent sequences: A196444 A196445 A196446 * A196448 A196449 A196450

Keyword

nonn,fini,full

Author

Nicolay Beluchenko, Oct 13 2011

Extensions

Edited by Nathaniel Johnston, May 11 2016

Status

approved