A056613 -id:A056613 - cp's OEIS frontend

A019473 Number of stable n-celled patterns ("still lifes") in Conway's Game of Life, up to rotation and reflection.

0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243, 273188, 672172, 1646147, 4051732, 9971377, 24619307, 60823008, 150613157, 373188952, 926068847, 2299616637, 5716948683, 14223867298, 35422864104

Offset: 1

Robert Munafo

This sequence only counts still lifes that cannot be broken down into 2 or more smaller still lifes. That is, it only counts "strict" still lifes (contrast with A056613). - Nathaniel Johnston, Dec 11 2019

			a(4)=2 because the block and the tub are the only 4-cell still lifes.

S. Ekström, Enumerating Still Lifes (in C)
N. D. Elkies, The still-Life density problem and its generalizations, arXiv:math/9905194 [math.CO], 1999.
H. Koenig, Stable Objects (another version of this sequence)
R. Munafo, Still-Lifes with N cells in Conway's game of Life
Mark D. Niemiec, Life Object Counts
R. C. Schroeppel, A few mathematical experiments
Eric Weisstein's World of Mathematics, Game of Life

Cf. A056613, A056614, A056605, A330283.

More terms from Stephen A. Silver, Dec 11 1999
a(24) corrected, at the suggestion of Mark Niemiec, by Nathaniel Johnston, Aug 26 2016
a(24)-a(28) corrected, using data computed by Simon Ekström, by Adam P. Goucher, Jan 08 2017
a(31)-a(32) from Nathaniel Johnston, using a script made by Simon Ekström, May 25 2017
a(33) from Nathaniel Johnston, using a script made by Simon Ekström, Apr 05 2019
a(34) from Nathaniel Johnston, using a script made by Simon Ekström, Jan 09 2020

A330283 Number of n-celled quasi still lifes in Conway's Game of Life, up to rotation and reflection.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 13, 57, 141, 465, 1224, 3956, 11599, 36538, 107415, 327250, 972040, 2957488, 8879327, 26943317

Offset: 1

Nathaniel Johnston, Dec 09 2019

A quasi still life is a still life made up of two or more strict still lifes (A019473) that share at least 1 neighboring dead cell, but those cells still remain dead due to underpopulation just like if we were to separate those component strict still lifes. Contrast with pseudo still lifes (A056613), where the component strict still lifes share at least 1 neighboring dead cell that remains dead, but the reason shifts from underpopulation in the strict still lifes to overpopulation in the combined pattern.

			When n = 8, the 6 quasi still lifes are the various arrangements of blocks and tubs in which they share a neighbor, but that neighbor remains dead due to underpopulation. In the diagrams below, "." is a dead cell, "o" is a live cell, and "*" is a dead neighboring cell that makes the pattern a quasi still life:
.o...o.
o.o*o.o
.o...o.
.
.o.....
o.o*.o.
.o.*o.o
.....o.
.
oo...
oo...
..*..
...oo
...oo
.
oo....
oo....
..*.o.
...o.o
....o.
.
.o.....
o.o....
.o.*.o.
....o.o
.....o.
.
.o....
o.o...
.o.*..
..*.o.
...o.o
....o.

LifeWiki, quasi still lifes

Cf. A019473, A056613.

A110910 Configurations in the evolution of a line of n cells in Conway's Game of Life, with 0=infinity. For periodic evolutions, a(n)=(preperiod length)+(period length). For non-periodic evolutions, a(n)=0.

Original entry on oeis.org

1, 2, 2, 2, 3, 8, 13, 15, 49, 22, 17, 17, 16, 26, 29, 41, 34, 25, 21, 26, 21, 21, 36, 31, 29, 95, 25, 29, 34, 38, 105, 150, 61, 582, 43, 58, 92, 108, 263, 277, 50, 212, 59, 53, 57, 99, 55, 170, 196, 812, 105, 54, 53, 85, 59, 81, 0, 418, 63, 63, 314, 117, 118, 170, 236, 104

Offset: 0

Paul Stoeber (pstoeber(AT)uni-potsdam.de), Oct 03 2005

If nothing catches up with an outbound glider, then a(n)=0 for n>=1000 because when you watch the horizontal 1000-line evolve in a simulator, around the 490th generation, gliders fly away from the left and right corners before the non-chaotic growing in the middle has finished, so you will see the same local picture in the 490th generation of longer lines.

			a(0)=1 because there is only the empty configuration. a(10)=2+15 because the 10-line needs two steps to become a pentadecathlon. a(56)=0 because the 56-line sends four gliders to outer space.

Berlekamp/Conway/Guy, Winning Ways ..., 2nd ed, vol. 4, chapter 25

Cf. A061342, A019473, A056605, A056614, A055397, A099733, A089520, A098720, A056613.

{- program for verification of periodic cases. The non-periodic cases listed here evolve into a periodic kernel plus gliders whose paths ahead do not intersect each other or the kernel (gliders marching in single file are not counted as intersecting). -}
import Data.Set
main = print [if n `elem` known then 0 else a n | n<-[0..105]]
known = [56, 71, 72, 75, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 96, 98, 100, 102, 103, 105]
a n = count empty (iterate evolve (fromList [(x, 0) | x<-[1..n]]))
neighbors (x, y) = fromList
                  [(x+u, y+v) | u<-[ -1, 0, 1], v<-[ -1, 0, 1], (u, v)/=(0, 0)]
evolve life =
  let fil f = Data.Set.filter
              (\x-> f (size (life `intersection` neighbors x)))
  in (life `difference` fil (\k-> k<2 || k>3) life) `union` fil (== 3)
     (unions (Prelude.map neighbors (elems life)) `difference` life)
count o (x:xs) | x `member` o = 0
               | otherwise = 1 + count (o `union` singleton x) xs

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A019473 Number of stable n-celled patterns ("still lifes") in Conway's Game of Life, up to rotation and reflection.

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A330283 Number of n-celled quasi still lifes in Conway's Game of Life, up to rotation and reflection.

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A110910 Configurations in the evolution of a line of n cells in Conway's Game of Life, with 0=infinity. For periodic evolutions, a(n)=(preperiod length)+(period length). For non-periodic evolutions, a(n)=0.

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