A056614 -id:A056614 - 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

A318858 Number of still-lifes synthesizable from n gliders in Conway's game of Life.

Original entry on oeis.org

0, 6, 11, 111, 114, 217

Offset: 1

Ed Pegg Jr, Sep 04 2018

No stable form can be produced by a single glider (on an infinite or toroidal board), thus a(1) = 0.
Block, boat, beehive, loaf, eater, and pond are the six still-life forms that can be made from 2 gliders.
Tub, Ship, Barge, Long boat, Mango, Long barge, Very long boat, Hat, Half bakery, Paperclip, and Bi-pond can be made from 3 gliders.
Most of the terms in this sequence are conjectural and will likely increase over time. - Nathaniel Johnston, Dec 03 2019

Mark Niemiec, Life Objects Sorted by Cost in Gliders.
Wikipedia, Glider (Conway's Life), Still life (cellular automaton)
Index entries for sequences related to cellular automata

Cf. A019473, A056614.

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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A318858 Number of still-lifes synthesizable from n gliders in Conway's game of Life.

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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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Haskell