login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A269712 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood. 0
1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Initialized with a single black (ON) cell at stage zero.

Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692  and 700 also generate this sequence.

Apparently a duplicate of A173033. - R. J. Mathar, Mar 09 2016

REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

LINKS

Table of n, a(n) for n=0..15.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

FORMULA

Conjectures from Colin Barker, Mar 08 2016: (Start)

a(n) = 4*(2^n-1) =A028399(n+2) for n>0.

a(n) = 3*a(n-1)-2*a(n-2) for n>2.

G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).

(End)

MATHEMATICA

rule=20; stages=300;

ca=CellularAutomaton[{rule, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, stages]; (* Start with single black cell *)

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

CROSSREFS

Cf. A269711.

Sequence in context: A261320 A179023 A321690 * A028399 A173033 A339124

Adjacent sequences:  A269709 A269710 A269711 * A269713 A269714 A269715

KEYWORD

nonn,more

AUTHOR

Robert Price, Mar 04 2016

EXTENSIONS

a(9)-a(15) from Lars Blomberg, Apr 15 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 15 10:49 EDT 2021. Contains 342977 sequences. (Running on oeis4.)