A019473 -id:A019473 - cp's OEIS frontend

A056613 Number of n-celled pseudo still lifes in Conway's Game of Life, up to rotation and reflection.

0, 0, 0, 0, 0, 0, 0, 1, 1, 7, 16, 55, 110, 279, 620, 1645, 4067, 10843, 27250, 70637, 179011, 462086, 1184882, 3069135, 7906676, 20463274, 52816265, 136655095, 353198379, 914075620, 2364815358, 6123084116, 15851861075, 41058173683

Offset: 1

N. J. A. Sloane, Aug 28 2000

There are two slightly different possible definitions for a pseudo still life: a still life that can be partitioned into exactly two different still lifes, or a still life that can be partitioned into two *or more* still lifes. This sequence uses the latter definition. The first point in the sequence where this makes a difference is a(32) = 6123084116, which would be a(32) = 6123084115 under the former definition. - Nathaniel Johnston, May 25 2017

			For n = 8, the unique pseudo still life is a pair of 2 X 2 blocks occupying a 5 X 2 bounding box.

S. Ekström, Enumerating Still Lifes (in C)
Mark D. Niemiec, Life Object Counts

Cf. A019473, A330283.

a(24) corrected by Nathaniel Johnston, Aug 26 2016 at the suggestion of Mark Niemiec
a(25)-a(30) computed by Simon Ekström and inserted by Adam P. Goucher, Jan 08 2017
a(24) corrected by Nathaniel Johnston, Feb 21 2017 (computed by Simon Ekström)
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

A056614 Number of n-celled P2-oscillators in Conway's game of Life.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 1, 1, 1, 1, 6, 3, 20, 29, 98, 199, 484, 1083, 2722, 6596

Offset: 1

N. J. A. Sloane, Aug 28 2000

The three six-bit period 2 oscillators are Toad, Clock, and Beacon. - Ed Pegg Jr, Sep 05 2018

Mark D. Niemiec, Life Object Counts
Mark D. Niemiec, Life Oscillators
Index entries for sequences related to cellular automata

Cf. A019473, A318858.

a(20)-a(21) from Ed Pegg Jr, Sep 05 2018

A089520 In Conway's Game of Life, the number of steps it takes for an n X n square, in which all the cells are in the "on" state, to die out or start to cycle, or -1 if there is no cycle.

Original entry on oeis.org

1, 0, 5, 4, 11, 5, 5, 6, 16, 17, 32, 9, 18, 9, 22, 11, 33, 17, 20, 12, 26, 13, 48, 15, 46, 26, 295, 45, 154, 38, 62, 309, 38, 87, 78, 53, 96, 150, 641, 69, 82, 265, 216, 70, 70, 70, 120, 401, 107, 78, 70, 351, 318, 109, 297, 95, 122, -1, -1, 85, 232, 294, 127

Offset: 1

Anne M. Donovan (anned3005(AT)aol.com), Nov 05 2003

sign

The -1 terms for n = 58, 59, 80, 92, 95, 96, 98, 99, 100 correspond to starting n x n squares that produce 8 gliders (16 for n = 99) that go off to infinity, hence never reaching a cycle. - Michael S. Branicky, Jul 06 2022

			a(1) = 1 since a single cell is switched off on step 1.
a(2) = 0 since a block is cyclic to start with: 0th = 1st generation.
a(3) = 5 since a cycle starts there: 5th = 7th generation.

Michael S. Branicky, Table of n, a(n) for n = 1..100
Michael S. Branicky, Python program and utilities.
D. Griffeath, Life32 by Johan Bontes.
Nathaniel Johnston, ConwayLife.com.
S. Silver, Life Lexicon.

Cf. A019473, A086993.

More terms from John W. Layman, Nov 07 2003
39 more terms from Rick L. Shepherd, Jun 04 2004
Name edited, a(58) and a(59) changed, and a(61) and beyond from Michael S. Branicky, Jul 06 2022

A166476 Number of stable n-celled patterns ("still lifes") in the 2x2 (B36/S125) cellular automaton.

Original entry on oeis.org

0, 2, 1, 3, 4, 9, 10, 27, 48, 126

Offset: 1

Nathaniel Johnston, Oct 14 2009

Counts only distinct strict still lifes; reflections and rotations are removed. Pseudo-still lifes (e.g., patterns made up of two disjoint still lifes) are not counted.

			a(3) = 1 because there is only one distinct still life with three cells: a diagonal line of length three.

LifeWiki, 2x2

Cf. A019473

A056605 Number of stable n-celled patterns ("still lifes") in Conway's game of Life (incorrect version).

Original entry on oeis.org

0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19046, 45766

Offset: 1

N. J. A. Sloane, Aug 28 2000

dead

a(18) and a(19) are incorrect. See A019473 for the correct version of this sequence. - Nathaniel Johnston, Aug 19 2018

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

Mark D. Niemiec, Life Object Counts
H. Koenig, Stable Objects

Cf. A019473.

A175000 Number of stable n-celled patterns ("still lifes") in the HighLife (B36/S23) cellular automaton.

Original entry on oeis.org

0, 0, 0, 2, 1, 4, 4, 9, 9, 25, 44, 111, 218

Offset: 1

Christian Schroeder, Apr 03 2010

Counts only distinct strict still lifes; reflections and rotations are removed. Pseudo-still lifes (e.g., patterns made up of two disjoint still lifes) are not counted.

			a(4)=2 because there are only two still-lives with four cells: the block (a 2x2 block) and the tub (four cells bordering all four sides of an empty spot).

LifeWiki, HighLife

Cf. A019473, A166476

A273308 Maximum population of a 2 X n still life in Conway's Game of Life.

Original entry on oeis.org

0, 4, 4, 6, 8, 8, 10, 12, 12, 14, 16, 16, 18, 20, 20, 22, 24, 24, 26, 28, 28, 30, 32, 32, 34, 36, 36, 38, 40, 40, 42, 44, 44, 46, 48, 48, 50, 52, 52, 54, 56, 56, 58, 60, 60, 62, 64, 64, 66, 68, 68, 70, 72, 72, 74, 76, 76, 78, 80, 80, 82, 84, 84, 86, 88, 88

Offset: 1

Nathaniel Johnston, May 19 2016

Although the Chu et al. reference does not discuss this problem explicitly, the same methods in that paper can be used to prove the formula for this sequence.

			a(2) = 4 because the largest number of alive cells in a 2 X 2 still life is 4, which is attained by the block.
a(4) = 6 because the largest number of alive cells in a 2 X 4 still life is 6, which is attained by the snake.

Colin Barker, Table of n, a(n) for n = 1..1000
G. Chu, P. Stuckey, and M.G. de la Banda, Using relaxations in Maximum Density Still Life, In Proc. of Fifteenth Intl. Conf. on Principles and Practice of Constraint Programming, 258-273 (2009).
LifeWiki, Still life
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A019473, A055397.

seq(4*floor((n+1)*(1/3))+2*floor((1/2)*(`mod`(n+1, 3))), n = 2 .. 110);

LinearRecurrence[{1,0,1,-1},{0,4,4,6,8},70] (* Harvey P. Dale, Apr 19 2023 *)

concat(0, Vec(2*x^2*(2+x^2-x^3)/((1-x)^2*(1+x+x^2)) + O(x^50))) \\ Colin Barker, May 24 2016

def A273308(n): return n+sum(divmod(n,3)) if n > 1 else 0 # Chai Wah Wu, Jan 29 2023

For n >= 1, a(3*n) = a(3*n-1) = 4*n and a(3*n+1) = 4*n+2.
From Colin Barker, May 24 2016: (Start)
a(n) = a(n-1)+a(n-3)-a(n-4) for n>5.
G.f.: 2*x^2*(2+x^2-x^3) / ((1-x)^2*(1+x+x^2)). (End)
a(n) = A063224(n+1) = A063200(n+1) for n>1. - R. J. Mathar, May 27 2016

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.

A156228 Number of lakes in Conway's Game of Life with 8*n cells.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 31, 98, 446, 1894, 9049, 43151

Offset: 1

Nathaniel Johnston, Feb 06 2009

a(n) is also the number of walks of length 4*n on a 2D lattice with the properties that: it turns 90 degrees after every step of length 1, it is a closed loop (i.e., it ends where it started) and it never crosses itself.

In A266549, the walks are allowed to continue straight ahead. - Pontus von Brömssen, May 06 2025

			a(2) = 0 because there are no lakes with 16 cells.

Nathaniel Johnston, Counting Lakes
LifeWiki, Lakes

Cf. A019473, A266549, A337550.

a(11) and a(12) added by Nathaniel Johnston, Mar 09 2009

A175001 Number of stable n-celled patterns ("still lifes") in the Move (a.k.a. Morley; B368/S245) cellular automaton.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 1, 3

Offset: 1

Christian Schroeder, Apr 03 2010

			a(4)=1 because there is only one still-life with four cells: the tub (four cells bordering all four sides of an empty spot).

LifeWiki, Move

Cf. A019473, A166476, A175000

cp's OEIS Frontend

A056613 Number of n-celled pseudo still lifes in Conway's Game of Life, up to rotation and reflection.

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A056614 Number of n-celled P2-oscillators in Conway's game of Life.

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A089520 In Conway's Game of Life, the number of steps it takes for an n X n square, in which all the cells are in the "on" state, to die out or start to cycle, or -1 if there is no cycle.

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A166476 Number of stable n-celled patterns ("still lifes") in the 2x2 (B36/S125) cellular automaton.

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A056605 Number of stable n-celled patterns ("still lifes") in Conway's game of Life (incorrect version).

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A175000 Number of stable n-celled patterns ("still lifes") in the HighLife (B36/S23) cellular automaton.

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A273308 Maximum population of a 2 X n still life in Conway's Game of Life.

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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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A156228 Number of lakes in Conway's Game of Life with 8*n cells.

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