

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.


2



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
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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 twocell 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)
LifeWiki, Garden of Eden
Wikipedia, Garden of Eden


EXAMPLE

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


CROSSREFS

Cf. A197734.
Sequence in context: A140798 A325022 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



