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!)
A273768 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 913", based on the 5-celled von Neumann neighborhood. 1
1, 5, 22, 70, 151, 272, 441, 666, 955, 1316, 1757, 2286, 2911, 3640, 4481, 5442, 6531, 7756, 9125, 10646, 12327, 14176, 16201, 18410, 20811, 23412, 26221, 29246, 32495, 35976, 39697, 43666, 47891, 52380, 57141, 62182, 67511, 73136, 79065, 85306, 91867, 98756 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

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

LINKS

Robert Price, Table of n, a(n) for n = 0..128

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, May 29 2016: (Start)

a(n) = (4*n^3+12*n^2+11*n-39)/3 for n>2.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>6.

G.f.: (1+x+8*x^2+8*x^3-16*x^4+5*x^5+x^6) / (1-x)^4.

(End)

MATHEMATICA

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=913; stages=128;

rule=IntegerDigits[code, 2, 10];

g=2*stages+1; (* Maximum size of grid *)

a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)

ca=a;

ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];

PrependTo[ca, a];

(* Trim full grid to reflect growth by one cell at each stage *)

k=(Length[ca[[1]]]+1)/2;

ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];

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

Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

CROSSREFS

Cf. A273766.

Sequence in context: A286711 A222632 A273336 * A032168 A246211 A000345

Adjacent sequences:  A273765 A273766 A273767 * A273769 A273770 A273771

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 29 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 January 27 09:01 EST 2020. Contains 331293 sequences. (Running on oeis4.)