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!)
A270087 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood. 4

%I

%S 1,4,5,40,0,121,0,225,0,361,0,529,0,729,0,961,0,1225,0,1521,0,1849,0,

%T 2209,0,2601,0,3025,0,3481,0,3969,0,4489,0,5041,0,5625,0,6241,0,6889,

%U 0,7569,0,8281,0,9025,0,9801,0,10609,0,11449,0,12321,0,13225,0

%N Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood.

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

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

%H Robert Price, <a href="/A270087/b270087.txt">Table of n, a(n) for n = 0..128</a>

%H Robert Price, <a href="/A270087/a270087.tmp.txt">Diagrams of the first 20 stages.</a>

%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%F Conjectures from _Colin Barker_, Mar 11 2016: (Start)

%F a(n) = -1/2*(-1+(-1)^n)*(1+2*n)^2 for n>3.

%F a(n) = 0 for n>3 and even.

%F a(n) = 4*n^2+4*n+1 for n>3 and odd.

%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>9.

%F G.f.: (1+4*x+2*x^2+28*x^3-12*x^4+13*x^5+14*x^6-22*x^7-5*x^8+9*x^9) / ((1-x)^3*(1+x)^3).

%F (End)

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

%t code=73; stages=128;

%t rule=IntegerDigits[code,2,10];

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

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

%t ca=a;

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

%t PrependTo[ca,a];

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

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

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

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

%K nonn,easy

%O 0,2

%A _Robert Price_, Mar 10 2016

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 May 17 12:01 EDT 2021. Contains 343971 sequences. (Running on oeis4.)