Sequence A019473: Still-Lifes with N cells in Conway's game of Life

This sequence, Sloane's A019473, gives the number of stable patterns (called "still-lifes") in Conway's game of Life that have N cells. The sequence runs:

0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243, 273188, 672172, 1646147, 4051711, ...

Here are illustrations of the first few terms:

A[1] = 0 ; A[2] = 0 ; A[3] = 0 A[4] = 2 o o o A[5] = 1 o o o o o o o o o o A[6] = 5 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 A[7] = 4 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

When N>=8, pseudo-still-lifes become possible. A pseudo-still-life has one-cell gaps separating connected parts, which are cllaed islands¹. Here is the smallest example, two blocks placed side by side. Such cases are not counted in sequence A019473:

o o o o o o o o

The empty cells between the islands can be influenced by the neighboring cells on both sides of the gap to make it a still-life when the pieces by themselves would not. Once of the islands is sometimes called an inductor. The smallest example of this is a 10-cell still-life called block on table :

o o o o o o o o o o

It is also possible to have a still-life that has multiple islands, and cannot be separated into two pieces, but can be separated into 3 or 4 pieces each of which is stable by itself. Here is the simplest known pattern of that type, a 32-cell pattern found by Gabriel Nivasch:

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

You can see a lot more pictures of still-lifes here.

Some other sequences are discussed here.

1 : Stephen Silver, Life Lexicon

This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2014 Dec 07.

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